TPTP Problem File: SLH0208^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Finite_Fields/0005_Ring_Characteristic/prob_00813_027374__18214332_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1558 ( 337 unt; 282 typ; 0 def)
% Number of atoms : 4342 ( 828 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 13286 ( 206 ~; 29 |; 522 &;10330 @)
% ( 0 <=>;2199 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg)
% Number of types : 34 ( 33 usr)
% Number of type conns : 1421 (1421 >; 0 *; 0 +; 0 <<)
% Number of symbols : 251 ( 249 usr; 7 con; 0-4 aty)
% Number of variables : 4278 ( 423 ^;3642 !; 213 ?;4278 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:21:40.700
%------------------------------------------------------------------------------
% Could-be-implicit typings (33)
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_set_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J_J,type,
set_li1882408696177261060omplex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J_J,type,
set_set_list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
list_list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
set_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
set_list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
set_set_complex: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_I_Eo_J_J,type,
list_list_o: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
set_list_o: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (249)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Complex__Ocomplex_J,type,
comple8424636186594484919omplex: set_set_complex > set_complex ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
comple2136024025076962759omplex: set_set_list_complex > set_list_complex ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
comple6939822128159878743list_a: set_set_list_list_a > set_list_list_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
comple8404747032580312297st_nat: set_set_list_nat > set_list_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
comple6928918032620976721list_a: set_set_list_a > set_list_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Finite__Set_OFpow_001_Eo,type,
finite_Fpow_o: set_o > set_set_o ).
thf(sy_c_Finite__Set_OFpow_001t__Complex__Ocomplex,type,
finite_Fpow_complex: set_complex > set_set_complex ).
thf(sy_c_Finite__Set_OFpow_001t__List__Olist_Itf__a_J,type,
finite_Fpow_list_a: set_list_a > set_set_list_a ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Nat__Onat_J,type,
finite_Fpow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_Ocard_001_Eo,type,
finite_card_o: set_o > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
finite_card_complex: set_complex > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_I_Eo_J,type,
finite_card_list_o: set_list_o > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Complex__Ocomplex_J,type,
finite5120063068150530198omplex: set_list_complex > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
finite5336269520247027750omplex: set_li1882408696177261060omplex > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
finite4595494376813527864list_a: set_list_list_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
finite7325466520557071688st_nat: set_list_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
finite9134805042761151410list_a: set_list_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
finite_card_list_a: set_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
finite_finite_o: set_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
finite_finite_list_o: set_list_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
finite8712137658972009173omplex: set_list_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
finite1660835950917165235list_a: set_list_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_Eo_J,type,
finite_finite_set_o: set_set_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
finite6551019134538273531omplex: set_set_complex > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
finite5282473924520328461list_a: set_set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Oinj__on_001_Eo_001_Eo,type,
inj_on_o_o: ( $o > $o ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001t__Nat__Onat,type,
inj_on_o_nat: ( $o > nat ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001tf__a,type,
inj_on_o_a: ( $o > a ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001t__Complex__Ocomplex_001_Eo,type,
inj_on_complex_o: ( complex > $o ) > set_complex > $o ).
thf(sy_c_Fun_Oinj__on_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
inj_on2498852929715845839omplex: ( complex > complex ) > set_complex > $o ).
thf(sy_c_Fun_Oinj__on_001t__Complex__Ocomplex_001t__Nat__Onat,type,
inj_on_complex_nat: ( complex > nat ) > set_complex > $o ).
thf(sy_c_Fun_Oinj__on_001t__Complex__Ocomplex_001tf__a,type,
inj_on_complex_a: ( complex > a ) > set_complex > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_Eo,type,
inj_on_nat_o: ( nat > $o ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Complex__Ocomplex,type,
inj_on_nat_complex: ( nat > complex ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_nat_set_nat: ( nat > set_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
inj_on_nat_a: ( nat > a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inj_on2776966659131765557et_nat: ( set_nat > set_set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001_Eo,type,
inj_on_a_o: ( a > $o ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Complex__Ocomplex,type,
inj_on_a_complex: ( a > complex ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
inj_on_a_nat: ( a > nat ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Nat__Onat,type,
groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
groups5521247699297860762_a_nat: ( list_a > nat ) > set_list_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Nat__Onat,type,
groups8758837469787661168ex_nat: ( set_complex > nat ) > set_set_complex > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J_001t__Nat__Onat,type,
groups6516131157293929088ex_nat: ( set_list_complex > nat ) > set_set_list_complex > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Nat__Onat,type,
groups2871842722561159296_a_nat: ( set_list_list_a > nat ) > set_set_list_list_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
groups7315335787803791778at_nat: ( set_list_nat > nat ) > set_set_list_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Nat__Onat,type,
groups5993734322560061562_a_nat: ( set_list_a > nat ) > set_set_list_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
groups6141743369313575924_a_nat: ( set_a > nat ) > set_set_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Nat__Onat,type,
groups6334556678337121940_a_nat: ( a > nat ) > set_a > nat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(sy_c_List_Ocount__list_001_Eo,type,
count_list_o: list_o > $o > nat ).
thf(sy_c_List_Ocount__list_001t__Complex__Ocomplex,type,
count_list_complex: list_complex > complex > nat ).
thf(sy_c_List_Ocount__list_001t__List__Olist_Itf__a_J,type,
count_list_list_a: list_list_a > list_a > nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Ocount__list_001tf__a,type,
count_list_a: list_a > a > nat ).
thf(sy_c_List_Odistinct_001_Eo,type,
distinct_o: list_o > $o ).
thf(sy_c_List_Odistinct_001t__Complex__Ocomplex,type,
distinct_complex: list_complex > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Complex__Ocomplex_J,type,
distin3828661287404608645omplex: list_list_complex > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
distinct_list_list_a: list_list_list_a > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_Itf__a_J,type,
distinct_list_a: list_list_a > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001_Eo,type,
linord3142498349692569832_set_o: set_o > list_o ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001_Eo,type,
set_o2: list_o > set_o ).
thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
set_complex2: list_complex > set_complex ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_I_Eo_J,type,
set_list_o2: list_list_o > set_list_o ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Complex__Ocomplex_J,type,
set_list_complex2: list_list_complex > set_list_complex ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
set_list_list_a2: list_list_list_a > set_list_list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_On__lists_001_Eo,type,
n_lists_o: nat > list_o > list_list_o ).
thf(sy_c_List_On__lists_001t__Complex__Ocomplex,type,
n_lists_complex: nat > list_complex > list_list_complex ).
thf(sy_c_List_On__lists_001t__List__Olist_Itf__a_J,type,
n_lists_list_a: nat > list_list_a > list_list_list_a ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Oproduct__lists_001t__Complex__Ocomplex,type,
produc7545014605101902079omplex: list_list_complex > list_list_complex ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_Itf__a_J,type,
product_lists_list_a: list_list_list_a > list_list_list_a ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Osubseqs_001_Eo,type,
subseqs_o: list_o > list_list_o ).
thf(sy_c_List_Osubseqs_001t__Complex__Ocomplex,type,
subseqs_complex: list_complex > list_list_complex ).
thf(sy_c_List_Osubseqs_001t__List__Olist_Itf__a_J,type,
subseqs_list_a: list_list_a > list_list_list_a ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
size_size_list_o: list_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
size_s7907857696548412130omplex: list_list_complex > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
size_s2403821588304063868list_a: list_list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Complex__Ocomplex_M_Eo_J,type,
ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
ord_less_eq_list_a_o: ( list_a > $o ) > ( list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
ord_le3922870914418331732omplex: set_list_complex > set_list_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
ord_le8488217952732425610list_a: set_list_list_a > set_list_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
ord_le4374716579403074808_set_o: set_set_o > set_set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
ord_le4750530260501030778omplex: set_set_complex > set_set_complex > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
ord_le8877086941679407844list_a: set_set_list_a > set_set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_I_Eo_J,type,
order_Greatest_set_o: ( set_o > $o ) > set_o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Complex__Ocomplex_J,type,
order_95770167153410891omplex: ( set_complex > $o ) > set_complex ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
order_733672244956367037list_a: ( set_list_a > $o ) > set_list_a ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_Itf__a_J,type,
order_Greatest_set_a: ( set_a > $o ) > set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
power_power_complex: complex > nat > complex ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
collect_list_o: ( list_o > $o ) > set_list_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
collect_list_complex: ( list_complex > $o ) > set_list_complex ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
collec1601192001008753443omplex: ( list_list_complex > $o ) > set_li1882408696177261060omplex ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
collec1292721268053437947list_a: ( list_list_list_a > $o ) > set_list_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_I_Eo_J,type,
collect_set_o: ( set_o > $o ) > set_set_o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Complex__Ocomplex,type,
image_o_complex: ( $o > complex ) > set_o > set_complex ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Complex__Ocomplex_J,type,
image_o_set_complex: ( $o > set_complex ) > set_o > set_set_complex ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_Itf__a_J,type,
image_o_set_a: ( $o > set_a ) > set_o > set_set_a ).
thf(sy_c_Set_Oimage_001_Eo_001tf__a,type,
image_o_a: ( $o > a ) > set_o > set_a ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001_Eo,type,
image_complex_o: ( complex > $o ) > set_complex > set_o ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
image_1468599708987790691omplex: ( complex > complex ) > set_complex > set_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Nat__Onat,type,
image_complex_nat: ( complex > nat ) > set_complex > set_nat ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Set__Oset_I_Eo_J,type,
image_complex_set_o: ( complex > set_o ) > set_complex > set_set_o ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Set__Oset_It__Complex__Ocomplex_J,type,
image_5702600179605932057omplex: ( complex > set_complex ) > set_complex > set_set_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Set__Oset_It__Nat__Onat_J,type,
image_6352962638927555131et_nat: ( complex > set_nat ) > set_complex > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Set__Oset_Itf__a_J,type,
image_complex_set_a: ( complex > set_a ) > set_complex > set_set_a ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001tf__a,type,
image_complex_a: ( complex > a ) > set_complex > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_list_o_set_o: ( list_o > set_o ) > set_list_o > set_set_o ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
image_1532835712220217129omplex: ( list_complex > set_complex ) > set_list_complex > set_set_complex ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_432481560377026271list_a: ( list_list_a > set_list_a ) > set_list_list_a > set_set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001tf__a,type,
image_list_nat_a: ( list_nat > a ) > set_list_nat > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001_Eo,type,
image_list_a_o: ( list_a > $o ) > set_list_a > set_o ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
image_list_a_nat: ( list_a > nat ) > set_list_a > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_list_a_set_nat: ( list_a > set_nat ) > set_list_a > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_list_a_set_a: ( list_a > set_a ) > set_list_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Complex__Ocomplex,type,
image_nat_complex: ( nat > complex ) > set_nat > set_complex ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Complex__Ocomplex_J,type,
image_6594795319511438139omplex: ( nat > set_complex ) > set_nat > set_set_complex ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
image_nat_set_a: ( nat > set_a ) > set_nat > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Complex__Ocomplex,type,
image_a_complex: ( a > complex ) > set_a > set_complex ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_I_Eo_J,type,
image_a_set_o: ( a > set_o ) > set_a > set_set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Complex__Ocomplex_J,type,
image_a_set_complex: ( a > set_complex ) > set_a > set_set_complex ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_a_set_list_a: ( a > set_list_a ) > set_a > set_set_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
image_a_set_nat: ( a > set_nat ) > set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_Eo,type,
set_ord_atMost_o: $o > set_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_I_Eo_J,type,
set_ord_atMost_set_o: set_o > set_set_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Complex__Ocomplex_J,type,
set_or9043709113427266269omplex: set_complex > set_set_complex ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_or6279072120763780779list_a: set_list_a > set_set_list_a ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__List__Olist_It__Complex__Ocomplex_J,type,
member_list_complex: list_complex > set_list_complex > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Complex__Ocomplex_J,type,
member_set_complex: set_complex > set_set_complex > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: set_a ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1275)
thf(fact_0_True,axiom,
finite_finite_a @ a2 ).
% True
thf(fact_1_card__lists__length__eq,axiom,
! [A: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( finite7325466520557071688st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A )
& ( ( size_s3023201423986296836st_nat @ Xs )
= N ) ) ) )
= ( power_power_nat @ ( finite_card_list_nat @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_2_card__lists__length__eq,axiom,
! [A: set_list_complex,N: nat] :
( ( finite8712137658972009173omplex @ A )
=> ( ( finite5336269520247027750omplex
@ ( collec1601192001008753443omplex
@ ^ [Xs: list_list_complex] :
( ( ord_le3922870914418331732omplex @ ( set_list_complex2 @ Xs ) @ A )
& ( ( size_s7907857696548412130omplex @ Xs )
= N ) ) ) )
= ( power_power_nat @ ( finite5120063068150530198omplex @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_3_card__lists__length__eq,axiom,
! [A: set_list_list_a,N: nat] :
( ( finite1660835950917165235list_a @ A )
=> ( ( finite4595494376813527864list_a
@ ( collec1292721268053437947list_a
@ ^ [Xs: list_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Xs ) @ A )
& ( ( size_s2403821588304063868list_a @ Xs )
= N ) ) ) )
= ( power_power_nat @ ( finite9134805042761151410list_a @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_4_card__lists__length__eq,axiom,
! [A: set_o,N: nat] :
( ( finite_finite_o @ A )
=> ( ( finite_card_list_o
@ ( collect_list_o
@ ^ [Xs: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A )
& ( ( size_size_list_o @ Xs )
= N ) ) ) )
= ( power_power_nat @ ( finite_card_o @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_5_card__lists__length__eq,axiom,
! [A: set_list_a,N: nat] :
( ( finite_finite_list_a @ A )
=> ( ( finite9134805042761151410list_a
@ ( collect_list_list_a
@ ^ [Xs: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A )
& ( ( size_s349497388124573686list_a @ Xs )
= N ) ) ) )
= ( power_power_nat @ ( finite_card_list_a @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_6_card__lists__length__eq,axiom,
! [A: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite5120063068150530198omplex
@ ( collect_list_complex
@ ^ [Xs: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A )
& ( ( size_s3451745648224563538omplex @ Xs )
= N ) ) ) )
= ( power_power_nat @ ( finite_card_complex @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_7_card__lists__length__eq,axiom,
! [A: set_a,N: nat] :
( ( finite_finite_a @ A )
=> ( ( finite_card_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A )
& ( ( size_size_list_a @ Xs )
= N ) ) ) )
= ( power_power_nat @ ( finite_card_a @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_8_card__lists__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
& ( ( size_size_list_nat @ Xs )
= N ) ) ) )
= ( power_power_nat @ ( finite_card_nat @ A ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_9_subsetI,axiom,
! [A: set_complex,B: set_complex] :
( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( member_complex @ X @ B ) )
=> ( ord_le211207098394363844omplex @ A @ B ) ) ).
% subsetI
thf(fact_10_subsetI,axiom,
! [A: set_list_a,B: set_list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ( member_list_a @ X @ B ) )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% subsetI
thf(fact_11_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ X @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_12_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ X @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_13_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_14_subset__antisym,axiom,
! [A: set_complex,B: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_le211207098394363844omplex @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_15_subset__antisym,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_16_subset__antisym,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_17_subset__antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_18_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_19_order__refl,axiom,
! [X2: set_complex] : ( ord_le211207098394363844omplex @ X2 @ X2 ) ).
% order_refl
thf(fact_20_order__refl,axiom,
! [X2: set_list_a] : ( ord_le8861187494160871172list_a @ X2 @ X2 ) ).
% order_refl
thf(fact_21_order__refl,axiom,
! [X2: set_o] : ( ord_less_eq_set_o @ X2 @ X2 ) ).
% order_refl
thf(fact_22_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_23_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_24_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_25_dual__order_Orefl,axiom,
! [A2: set_complex] : ( ord_le211207098394363844omplex @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_26_dual__order_Orefl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_27_dual__order_Orefl,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_28_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_29_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_30_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_31_subset__code_I1_J,axiom,
! [Xs2: list_complex,B: set_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
=> ( member_complex @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_32_subset__code_I1_J,axiom,
! [Xs2: list_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ B )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs2 ) )
=> ( member_list_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_33_subset__code_I1_J,axiom,
! [Xs2: list_o,B: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ B )
= ( ! [X3: $o] :
( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
=> ( member_o @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_34_subset__code_I1_J,axiom,
! [Xs2: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
=> ( member_a @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_35_subset__code_I1_J,axiom,
! [Xs2: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_36_Collect__subset,axiom,
! [A: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_37_Collect__subset,axiom,
! [A: set_list_complex,P: list_complex > $o] :
( ord_le3922870914418331732omplex
@ ( collect_list_complex
@ ^ [X3: list_complex] :
( ( member_list_complex @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_38_Collect__subset,axiom,
! [A: set_list_list_a,P: list_list_a > $o] :
( ord_le8488217952732425610list_a
@ ( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_39_Collect__subset,axiom,
! [A: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_40_Collect__subset,axiom,
! [A: set_list_a,P: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_41_Collect__subset,axiom,
! [A: set_complex,P: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_42_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_43_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) )
@ A ) ).
% Collect_subset
thf(fact_44_conj__subset__def,axiom,
! [A: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ A
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le6045566169113846134st_nat @ A @ ( collect_list_nat @ P ) )
& ( ord_le6045566169113846134st_nat @ A @ ( collect_list_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_45_conj__subset__def,axiom,
! [A: set_list_complex,P: list_complex > $o,Q: list_complex > $o] :
( ( ord_le3922870914418331732omplex @ A
@ ( collect_list_complex
@ ^ [X3: list_complex] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le3922870914418331732omplex @ A @ ( collect_list_complex @ P ) )
& ( ord_le3922870914418331732omplex @ A @ ( collect_list_complex @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_46_conj__subset__def,axiom,
! [A: set_list_list_a,P: list_list_a > $o,Q: list_list_a > $o] :
( ( ord_le8488217952732425610list_a @ A
@ ( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le8488217952732425610list_a @ A @ ( collect_list_list_a @ P ) )
& ( ord_le8488217952732425610list_a @ A @ ( collect_list_list_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_47_conj__subset__def,axiom,
! [A: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A
@ ( collect_o
@ ^ [X3: $o] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_o @ A @ ( collect_o @ P ) )
& ( ord_less_eq_set_o @ A @ ( collect_o @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_48_conj__subset__def,axiom,
! [A: set_list_a,P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le8861187494160871172list_a @ A @ ( collect_list_a @ P ) )
& ( ord_le8861187494160871172list_a @ A @ ( collect_list_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_49_conj__subset__def,axiom,
! [A: set_complex,P: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ A
@ ( collect_complex
@ ^ [X3: complex] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_le211207098394363844omplex @ A @ ( collect_complex @ P ) )
& ( ord_le211207098394363844omplex @ A @ ( collect_complex @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_50_conj__subset__def,axiom,
! [A: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_a @ A @ ( collect_a @ P ) )
& ( ord_less_eq_set_a @ A @ ( collect_a @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_51_conj__subset__def,axiom,
! [A: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_52_prop__restrict,axiom,
! [X2: list_nat,Z: set_list_nat,X4: set_list_nat,P: list_nat > $o] :
( ( member_list_nat @ X2 @ Z )
=> ( ( ord_le6045566169113846134st_nat @ Z
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_53_prop__restrict,axiom,
! [X2: list_complex,Z: set_list_complex,X4: set_list_complex,P: list_complex > $o] :
( ( member_list_complex @ X2 @ Z )
=> ( ( ord_le3922870914418331732omplex @ Z
@ ( collect_list_complex
@ ^ [X3: list_complex] :
( ( member_list_complex @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_54_prop__restrict,axiom,
! [X2: list_list_a,Z: set_list_list_a,X4: set_list_list_a,P: list_list_a > $o] :
( ( member_list_list_a @ X2 @ Z )
=> ( ( ord_le8488217952732425610list_a @ Z
@ ( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( member_list_list_a @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_55_prop__restrict,axiom,
! [X2: $o,Z: set_o,X4: set_o,P: $o > $o] :
( ( member_o @ X2 @ Z )
=> ( ( ord_less_eq_set_o @ Z
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_56_prop__restrict,axiom,
! [X2: list_a,Z: set_list_a,X4: set_list_a,P: list_a > $o] :
( ( member_list_a @ X2 @ Z )
=> ( ( ord_le8861187494160871172list_a @ Z
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_57_prop__restrict,axiom,
! [X2: complex,Z: set_complex,X4: set_complex,P: complex > $o] :
( ( member_complex @ X2 @ Z )
=> ( ( ord_le211207098394363844omplex @ Z
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_58_prop__restrict,axiom,
! [X2: a,Z: set_a,X4: set_a,P: a > $o] :
( ( member_a @ X2 @ Z )
=> ( ( ord_less_eq_set_a @ Z
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_59_prop__restrict,axiom,
! [X2: nat,Z: set_nat,X4: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ Z )
=> ( ( ord_less_eq_set_nat @ Z
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_60_Collect__restrict,axiom,
! [X4: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_61_Collect__restrict,axiom,
! [X4: set_list_complex,P: list_complex > $o] :
( ord_le3922870914418331732omplex
@ ( collect_list_complex
@ ^ [X3: list_complex] :
( ( member_list_complex @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_62_Collect__restrict,axiom,
! [X4: set_list_list_a,P: list_list_a > $o] :
( ord_le8488217952732425610list_a
@ ( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( member_list_list_a @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_63_Collect__restrict,axiom,
! [X4: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_64_Collect__restrict,axiom,
! [X4: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_65_Collect__restrict,axiom,
! [X4: set_complex,P: complex > $o] :
( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_66_Collect__restrict,axiom,
! [X4: set_list_a,P: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_67_Collect__restrict,axiom,
! [X4: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_68_subset__CollectI,axiom,
! [B: set_list_nat,A: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ! [X: list_nat] :
( ( member_list_nat @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_69_subset__CollectI,axiom,
! [B: set_list_complex,A: set_list_complex,Q: list_complex > $o,P: list_complex > $o] :
( ( ord_le3922870914418331732omplex @ B @ A )
=> ( ! [X: list_complex] :
( ( member_list_complex @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le3922870914418331732omplex
@ ( collect_list_complex
@ ^ [X3: list_complex] :
( ( member_list_complex @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_list_complex
@ ^ [X3: list_complex] :
( ( member_list_complex @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_70_subset__CollectI,axiom,
! [B: set_list_list_a,A: set_list_list_a,Q: list_list_a > $o,P: list_list_a > $o] :
( ( ord_le8488217952732425610list_a @ B @ A )
=> ( ! [X: list_list_a] :
( ( member_list_list_a @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le8488217952732425610list_a
@ ( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( member_list_list_a @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_71_subset__CollectI,axiom,
! [B: set_a,A: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_72_subset__CollectI,axiom,
! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_73_subset__CollectI,axiom,
! [B: set_complex,A: set_complex,Q: complex > $o,P: complex > $o] :
( ( ord_le211207098394363844omplex @ B @ A )
=> ( ! [X: complex] :
( ( member_complex @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le211207098394363844omplex
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_74_subset__CollectI,axiom,
! [B: set_list_a,A: set_list_a,Q: list_a > $o,P: list_a > $o] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_75_subset__CollectI,axiom,
! [B: set_o,A: set_o,Q: $o > $o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_o
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ B )
& ( Q @ X3 ) ) )
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_76_subset__Collect__iff,axiom,
! [B: set_list_nat,A: set_list_nat,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ B
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_77_subset__Collect__iff,axiom,
! [B: set_list_complex,A: set_list_complex,P: list_complex > $o] :
( ( ord_le3922870914418331732omplex @ B @ A )
=> ( ( ord_le3922870914418331732omplex @ B
@ ( collect_list_complex
@ ^ [X3: list_complex] :
( ( member_list_complex @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: list_complex] :
( ( member_list_complex @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_78_subset__Collect__iff,axiom,
! [B: set_list_list_a,A: set_list_list_a,P: list_list_a > $o] :
( ( ord_le8488217952732425610list_a @ B @ A )
=> ( ( ord_le8488217952732425610list_a @ B
@ ( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_79_subset__Collect__iff,axiom,
! [B: set_a,A: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ B
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_80_subset__Collect__iff,axiom,
! [B: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_81_subset__Collect__iff,axiom,
! [B: set_complex,A: set_complex,P: complex > $o] :
( ( ord_le211207098394363844omplex @ B @ A )
=> ( ( ord_le211207098394363844omplex @ B
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_82_subset__Collect__iff,axiom,
! [B: set_list_a,A: set_list_a,P: list_a > $o] :
( ( ord_le8861187494160871172list_a @ B @ A )
=> ( ( ord_le8861187494160871172list_a @ B
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( member_list_a @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_83_subset__Collect__iff,axiom,
! [B: set_o,A: set_o,P: $o > $o] :
( ( ord_less_eq_set_o @ B @ A )
=> ( ( ord_less_eq_set_o @ B
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ X3 ) ) ) )
= ( ! [X3: $o] :
( ( member_o @ X3 @ B )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_84_List_Ofinite__set,axiom,
! [Xs2: list_o] : ( finite_finite_o @ ( set_o2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_85_List_Ofinite__set,axiom,
! [Xs2: list_a] : ( finite_finite_a @ ( set_a2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_86_List_Ofinite__set,axiom,
! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_87_List_Ofinite__set,axiom,
! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_88_List_Ofinite__set,axiom,
! [Xs2: list_list_a] : ( finite_finite_list_a @ ( set_list_a2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_89_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A3 )
@ ^ [X3: a] : ( member_a @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_90_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_91_less__eq__set__def,axiom,
( ord_le211207098394363844omplex
= ( ^ [A3: set_complex,B2: set_complex] :
( ord_le4573692005234683329plex_o
@ ^ [X3: complex] : ( member_complex @ X3 @ A3 )
@ ^ [X3: complex] : ( member_complex @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_92_less__eq__set__def,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B2: set_list_a] :
( ord_less_eq_list_a_o
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A3 )
@ ^ [X3: list_a] : ( member_list_a @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_93_less__eq__set__def,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
( ord_less_eq_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ A3 )
@ ^ [X3: $o] : ( member_o @ X3 @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_94_finite__lists__length__le,axiom,
! [A: set_a,N: nat] :
( ( finite_finite_a @ A )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_95_finite__lists__length__le,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_96_finite__lists__length__le,axiom,
! [A: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_97_finite__lists__length__le,axiom,
! [A: set_list_a,N: nat] :
( ( finite_finite_list_a @ A )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [Xs: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_98_finite__lists__length__le,axiom,
! [A: set_o,N: nat] :
( ( finite_finite_o @ A )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_99_finite__list,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ? [Xs3: list_o] :
( ( set_o2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_100_finite__list,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ? [Xs3: list_a] :
( ( set_a2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_101_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_102_finite__list,axiom,
! [A: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ? [Xs3: list_complex] :
( ( set_complex2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_103_finite__list,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ? [Xs3: list_list_a] :
( ( set_list_a2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_104_finite__lists__length__eq,axiom,
! [A: set_a,N: nat] :
( ( finite_finite_a @ A )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A )
& ( ( size_size_list_a @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_105_finite__lists__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
& ( ( size_size_list_nat @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_106_finite__lists__length__eq,axiom,
! [A: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A )
& ( ( size_s3451745648224563538omplex @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_107_finite__lists__length__eq,axiom,
! [A: set_list_a,N: nat] :
( ( finite_finite_list_a @ A )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [Xs: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A )
& ( ( size_s349497388124573686list_a @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_108_finite__lists__length__eq,axiom,
! [A: set_o,N: nat] :
( ( finite_finite_o @ A )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A )
& ( ( size_size_list_o @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_109_card__length,axiom,
! [Xs2: list_o] : ( ord_less_eq_nat @ ( finite_card_o @ ( set_o2 @ Xs2 ) ) @ ( size_size_list_o @ Xs2 ) ) ).
% card_length
thf(fact_110_card__length,axiom,
! [Xs2: list_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( set_list_nat2 @ Xs2 ) ) @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% card_length
thf(fact_111_card__length,axiom,
! [Xs2: list_list_complex] : ( ord_less_eq_nat @ ( finite5120063068150530198omplex @ ( set_list_complex2 @ Xs2 ) ) @ ( size_s7907857696548412130omplex @ Xs2 ) ) ).
% card_length
thf(fact_112_card__length,axiom,
! [Xs2: list_list_list_a] : ( ord_less_eq_nat @ ( finite9134805042761151410list_a @ ( set_list_list_a2 @ Xs2 ) ) @ ( size_s2403821588304063868list_a @ Xs2 ) ) ).
% card_length
thf(fact_113_card__length,axiom,
! [Xs2: list_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( set_a2 @ Xs2 ) ) @ ( size_size_list_a @ Xs2 ) ) ).
% card_length
thf(fact_114_card__length,axiom,
! [Xs2: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% card_length
thf(fact_115_card__length,axiom,
! [Xs2: list_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( set_complex2 @ Xs2 ) ) @ ( size_s3451745648224563538omplex @ Xs2 ) ) ).
% card_length
thf(fact_116_card__length,axiom,
! [Xs2: list_list_a] : ( ord_less_eq_nat @ ( finite_card_list_a @ ( set_list_a2 @ Xs2 ) ) @ ( size_s349497388124573686list_a @ Xs2 ) ) ).
% card_length
thf(fact_117_order__antisym__conv,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_118_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_119_order__antisym__conv,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_120_order__antisym__conv,axiom,
! [Y: set_complex,X2: set_complex] :
( ( ord_le211207098394363844omplex @ Y @ X2 )
=> ( ( ord_le211207098394363844omplex @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_121_order__antisym__conv,axiom,
! [Y: set_list_a,X2: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X2 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_122_order__antisym__conv,axiom,
! [Y: set_o,X2: set_o] :
( ( ord_less_eq_set_o @ Y @ X2 )
=> ( ( ord_less_eq_set_o @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_123_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_124_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_125_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_126_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_127_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_128_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_complex,C: set_complex] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_129_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_o,C: set_o] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_130_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_131_ord__le__eq__subst,axiom,
! [A2: set_complex,B3: set_complex,F: set_complex > nat,C: nat] :
( ( ord_le211207098394363844omplex @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_132_ord__le__eq__subst,axiom,
! [A2: set_o,B3: set_o,F: set_o > nat,C: nat] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_o,Y2: set_o] :
( ( ord_less_eq_set_o @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_133_ord__le__eq__subst,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_134_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_135_ord__eq__le__subst,axiom,
! [A2: nat,F: set_a > nat,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_136_ord__eq__le__subst,axiom,
! [A2: set_a,F: nat > set_a,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_137_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_138_ord__eq__le__subst,axiom,
! [A2: set_complex,F: nat > set_complex,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_139_ord__eq__le__subst,axiom,
! [A2: set_o,F: nat > set_o,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_140_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_141_ord__eq__le__subst,axiom,
! [A2: nat,F: set_complex > nat,B3: set_complex,C: set_complex] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_le211207098394363844omplex @ B3 @ C )
=> ( ! [X: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_142_ord__eq__le__subst,axiom,
! [A2: nat,F: set_o > nat,B3: set_o,C: set_o] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_o @ B3 @ C )
=> ( ! [X: set_o,Y2: set_o] :
( ( ord_less_eq_set_o @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_143_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_144_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_145_order__eq__refl,axiom,
! [X2: set_a,Y: set_a] :
( ( X2 = Y )
=> ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_146_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_147_order__eq__refl,axiom,
! [X2: set_nat,Y: set_nat] :
( ( X2 = Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_148_order__eq__refl,axiom,
! [X2: set_complex,Y: set_complex] :
( ( X2 = Y )
=> ( ord_le211207098394363844omplex @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_149_order__eq__refl,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( X2 = Y )
=> ( ord_le8861187494160871172list_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_150_order__eq__refl,axiom,
! [X2: set_o,Y: set_o] :
( ( X2 = Y )
=> ( ord_less_eq_set_o @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_151_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_152_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_153_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_154_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_155_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_complex,C: set_complex] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_le211207098394363844omplex @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_156_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_o,C: set_o] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_o @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_157_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_158_order__subst2,axiom,
! [A2: set_complex,B3: set_complex,F: set_complex > nat,C: nat] :
( ( ord_le211207098394363844omplex @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_159_order__subst2,axiom,
! [A2: set_o,B3: set_o,F: set_o > nat,C: nat] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: set_o,Y2: set_o] :
( ( ord_less_eq_set_o @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_160_order__subst2,axiom,
! [A2: set_a,B3: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_161_order__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_162_order__subst1,axiom,
! [A2: set_a,F: nat > set_a,B3: nat,C: nat] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_163_order__subst1,axiom,
! [A2: nat,F: set_a > nat,B3: set_a,C: set_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_164_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_165_order__subst1,axiom,
! [A2: nat,F: set_complex > nat,B3: set_complex,C: set_complex] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_le211207098394363844omplex @ B3 @ C )
=> ( ! [X: set_complex,Y2: set_complex] :
( ( ord_le211207098394363844omplex @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_166_order__subst1,axiom,
! [A2: nat,F: set_o > nat,B3: set_o,C: set_o] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_o @ B3 @ C )
=> ( ! [X: set_o,Y2: set_o] :
( ( ord_less_eq_set_o @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_167_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_168_order__subst1,axiom,
! [A2: set_complex,F: nat > set_complex,B3: nat,C: nat] :
( ( ord_le211207098394363844omplex @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le211207098394363844omplex @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le211207098394363844omplex @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_169_order__subst1,axiom,
! [A2: set_o,F: nat > set_o,B3: nat,C: nat] :
( ( ord_less_eq_set_o @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_170_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ! [X: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_171_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_172_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_173_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_174_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_complex,Z2: set_complex] : ( Y3 = Z2 ) )
= ( ^ [A4: set_complex,B4: set_complex] :
( ( ord_le211207098394363844omplex @ A4 @ B4 )
& ( ord_le211207098394363844omplex @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_175_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_a,Z2: set_list_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_176_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_o,Z2: set_o] : ( Y3 = Z2 ) )
= ( ^ [A4: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ A4 @ B4 )
& ( ord_less_eq_set_o @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_177_antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_178_antisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_179_antisym,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_180_antisym,axiom,
! [A2: set_complex,B3: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B3 )
=> ( ( ord_le211207098394363844omplex @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_181_antisym,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_182_antisym,axiom,
! [A2: set_o,B3: set_o] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ord_less_eq_set_o @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_183_dual__order_Otrans,axiom,
! [B3: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B3 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_184_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_185_dual__order_Otrans,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_186_dual__order_Otrans,axiom,
! [B3: set_complex,A2: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ B3 @ A2 )
=> ( ( ord_le211207098394363844omplex @ C @ B3 )
=> ( ord_le211207098394363844omplex @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_187_dual__order_Otrans,axiom,
! [B3: set_list_a,A2: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ C @ B3 )
=> ( ord_le8861187494160871172list_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_188_dual__order_Otrans,axiom,
! [B3: set_o,A2: set_o,C: set_o] :
( ( ord_less_eq_set_o @ B3 @ A2 )
=> ( ( ord_less_eq_set_o @ C @ B3 )
=> ( ord_less_eq_set_o @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_189_dual__order_Oantisym,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_190_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_191_dual__order_Oantisym,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_192_dual__order_Oantisym,axiom,
! [B3: set_complex,A2: set_complex] :
( ( ord_le211207098394363844omplex @ B3 @ A2 )
=> ( ( ord_le211207098394363844omplex @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_193_dual__order_Oantisym,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_194_dual__order_Oantisym,axiom,
! [B3: set_o,A2: set_o] :
( ( ord_less_eq_set_o @ B3 @ A2 )
=> ( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_195_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_196_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_197_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_198_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_complex,Z2: set_complex] : ( Y3 = Z2 ) )
= ( ^ [A4: set_complex,B4: set_complex] :
( ( ord_le211207098394363844omplex @ B4 @ A4 )
& ( ord_le211207098394363844omplex @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_199_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_list_a,Z2: set_list_a] : ( Y3 = Z2 ) )
= ( ^ [A4: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B4 @ A4 )
& ( ord_le8861187494160871172list_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_200_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_o,Z2: set_o] : ( Y3 = Z2 ) )
= ( ^ [A4: set_o,B4: set_o] :
( ( ord_less_eq_set_o @ B4 @ A4 )
& ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_201_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_202_order__trans,axiom,
! [X2: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_203_order__trans,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_204_order__trans,axiom,
! [X2: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z3 )
=> ( ord_less_eq_set_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_205_order__trans,axiom,
! [X2: set_complex,Y: set_complex,Z3: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y )
=> ( ( ord_le211207098394363844omplex @ Y @ Z3 )
=> ( ord_le211207098394363844omplex @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_206_order__trans,axiom,
! [X2: set_list_a,Y: set_list_a,Z3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ Z3 )
=> ( ord_le8861187494160871172list_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_207_order__trans,axiom,
! [X2: set_o,Y: set_o,Z3: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y )
=> ( ( ord_less_eq_set_o @ Y @ Z3 )
=> ( ord_less_eq_set_o @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_208_order_Otrans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_209_order_Otrans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_210_order_Otrans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_211_order_Otrans,axiom,
! [A2: set_complex,B3: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B3 )
=> ( ( ord_le211207098394363844omplex @ B3 @ C )
=> ( ord_le211207098394363844omplex @ A2 @ C ) ) ) ).
% order.trans
thf(fact_212_order_Otrans,axiom,
! [A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C )
=> ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_213_order_Otrans,axiom,
! [A2: set_o,B3: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( ord_less_eq_set_o @ B3 @ C )
=> ( ord_less_eq_set_o @ A2 @ C ) ) ) ).
% order.trans
thf(fact_214_order__antisym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_215_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_216_order__antisym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_217_order__antisym,axiom,
! [X2: set_complex,Y: set_complex] :
( ( ord_le211207098394363844omplex @ X2 @ Y )
=> ( ( ord_le211207098394363844omplex @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_218_order__antisym,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y )
=> ( ( ord_le8861187494160871172list_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_219_order__antisym,axiom,
! [X2: set_o,Y: set_o] :
( ( ord_less_eq_set_o @ X2 @ Y )
=> ( ( ord_less_eq_set_o @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_220_ord__le__eq__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_221_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_222_ord__le__eq__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_223_ord__le__eq__trans,axiom,
! [A2: set_complex,B3: set_complex,C: set_complex] :
( ( ord_le211207098394363844omplex @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le211207098394363844omplex @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_224_ord__le__eq__trans,axiom,
! [A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_225_ord__le__eq__trans,axiom,
! [A2: set_o,B3: set_o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_o @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_226_ord__eq__le__trans,axiom,
! [A2: set_a,B3: set_a,C: set_a] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_227_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_228_ord__eq__le__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_229_ord__eq__le__trans,axiom,
! [A2: set_complex,B3: set_complex,C: set_complex] :
( ( A2 = B3 )
=> ( ( ord_le211207098394363844omplex @ B3 @ C )
=> ( ord_le211207098394363844omplex @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_230_ord__eq__le__trans,axiom,
! [A2: set_list_a,B3: set_list_a,C: set_list_a] :
( ( A2 = B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C )
=> ( ord_le8861187494160871172list_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_231_ord__eq__le__trans,axiom,
! [A2: set_o,B3: set_o,C: set_o] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_o @ B3 @ C )
=> ( ord_less_eq_set_o @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_232_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [X3: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_233_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_234_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_235_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_complex,Z2: set_complex] : ( Y3 = Z2 ) )
= ( ^ [X3: set_complex,Y4: set_complex] :
( ( ord_le211207098394363844omplex @ X3 @ Y4 )
& ( ord_le211207098394363844omplex @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_236_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_a,Z2: set_list_a] : ( Y3 = Z2 ) )
= ( ^ [X3: set_list_a,Y4: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y4 )
& ( ord_le8861187494160871172list_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_237_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_o,Z2: set_o] : ( Y3 = Z2 ) )
= ( ^ [X3: set_o,Y4: set_o] :
( ( ord_less_eq_set_o @ X3 @ Y4 )
& ( ord_less_eq_set_o @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_238_le__cases3,axiom,
! [X2: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_239_nle__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_240_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_241_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_242_mem__Collect__eq,axiom,
! [A2: complex,P: complex > $o] :
( ( member_complex @ A2 @ ( collect_complex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_243_mem__Collect__eq,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_244_mem__Collect__eq,axiom,
! [A2: list_complex,P: list_complex > $o] :
( ( member_list_complex @ A2 @ ( collect_list_complex @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_245_mem__Collect__eq,axiom,
! [A2: list_list_a,P: list_list_a > $o] :
( ( member_list_list_a @ A2 @ ( collect_list_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_246_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_247_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_248_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_249_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_250_Collect__mem__eq,axiom,
! [A: set_complex] :
( ( collect_complex
@ ^ [X3: complex] : ( member_complex @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_251_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_252_Collect__mem__eq,axiom,
! [A: set_list_complex] :
( ( collect_list_complex
@ ^ [X3: list_complex] : ( member_list_complex @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_253_Collect__mem__eq,axiom,
! [A: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_254_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_255_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X3: $o] : ( member_o @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_256_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X: list_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_257_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_258_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X: complex] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_259_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X: list_nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_260_Collect__cong,axiom,
! [P: list_complex > $o,Q: list_complex > $o] :
( ! [X: list_complex] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_list_complex @ P )
= ( collect_list_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_261_Collect__cong,axiom,
! [P: list_list_a > $o,Q: list_list_a > $o] :
( ! [X: list_list_a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_list_list_a @ P )
= ( collect_list_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_262_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_263_Collect__cong,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X: $o] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_o @ P )
= ( collect_o @ Q ) ) ) ).
% Collect_cong
thf(fact_264_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X3: list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_265_Collect__mono__iff,axiom,
! [P: list_complex > $o,Q: list_complex > $o] :
( ( ord_le3922870914418331732omplex @ ( collect_list_complex @ P ) @ ( collect_list_complex @ Q ) )
= ( ! [X3: list_complex] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_266_Collect__mono__iff,axiom,
! [P: list_list_a > $o,Q: list_list_a > $o] :
( ( ord_le8488217952732425610list_a @ ( collect_list_list_a @ P ) @ ( collect_list_list_a @ Q ) )
= ( ! [X3: list_list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_267_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_268_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_269_Collect__mono__iff,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
= ( ! [X3: complex] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_270_Collect__mono__iff,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
= ( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_271_Collect__mono__iff,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) )
= ( ! [X3: $o] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_272_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_273_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_274_set__eq__subset,axiom,
( ( ^ [Y3: set_complex,Z2: set_complex] : ( Y3 = Z2 ) )
= ( ^ [A3: set_complex,B2: set_complex] :
( ( ord_le211207098394363844omplex @ A3 @ B2 )
& ( ord_le211207098394363844omplex @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_275_set__eq__subset,axiom,
( ( ^ [Y3: set_list_a,Z2: set_list_a] : ( Y3 = Z2 ) )
= ( ^ [A3: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B2 )
& ( ord_le8861187494160871172list_a @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_276_set__eq__subset,axiom,
( ( ^ [Y3: set_o,Z2: set_o] : ( Y3 = Z2 ) )
= ( ^ [A3: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A3 @ B2 )
& ( ord_less_eq_set_o @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_277_subset__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_278_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_279_subset__trans,axiom,
! [A: set_complex,B: set_complex,C2: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_le211207098394363844omplex @ B @ C2 )
=> ( ord_le211207098394363844omplex @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_280_subset__trans,axiom,
! [A: set_list_a,B: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C2 )
=> ( ord_le8861187494160871172list_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_281_subset__trans,axiom,
! [A: set_o,B: set_o,C2: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_set_o @ B @ C2 )
=> ( ord_less_eq_set_o @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_282_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X: list_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_283_Collect__mono,axiom,
! [P: list_complex > $o,Q: list_complex > $o] :
( ! [X: list_complex] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le3922870914418331732omplex @ ( collect_list_complex @ P ) @ ( collect_list_complex @ Q ) ) ) ).
% Collect_mono
thf(fact_284_Collect__mono,axiom,
! [P: list_list_a > $o,Q: list_list_a > $o] :
( ! [X: list_list_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le8488217952732425610list_a @ ( collect_list_list_a @ P ) @ ( collect_list_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_285_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_286_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_287_Collect__mono,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X: complex] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% Collect_mono
thf(fact_288_Collect__mono,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X: list_a] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_289_Collect__mono,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X: $o] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_mono
thf(fact_290_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_291_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_292_subset__refl,axiom,
! [A: set_complex] : ( ord_le211207098394363844omplex @ A @ A ) ).
% subset_refl
thf(fact_293_subset__refl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% subset_refl
thf(fact_294_subset__refl,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ A ) ).
% subset_refl
thf(fact_295_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_296_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_297_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A3: set_complex,B2: set_complex] :
! [T: complex] :
( ( member_complex @ T @ A3 )
=> ( member_complex @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_298_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B2: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A3 )
=> ( member_list_a @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_299_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
! [T: $o] :
( ( member_o @ T @ A3 )
=> ( member_o @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_300_equalityD2,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% equalityD2
thf(fact_301_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_302_equalityD2,axiom,
! [A: set_complex,B: set_complex] :
( ( A = B )
=> ( ord_le211207098394363844omplex @ B @ A ) ) ).
% equalityD2
thf(fact_303_equalityD2,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ( ord_le8861187494160871172list_a @ B @ A ) ) ).
% equalityD2
thf(fact_304_equalityD2,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ( ord_less_eq_set_o @ B @ A ) ) ).
% equalityD2
thf(fact_305_equalityD1,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% equalityD1
thf(fact_306_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_307_equalityD1,axiom,
! [A: set_complex,B: set_complex] :
( ( A = B )
=> ( ord_le211207098394363844omplex @ A @ B ) ) ).
% equalityD1
thf(fact_308_equalityD1,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ( ord_le8861187494160871172list_a @ A @ B ) ) ).
% equalityD1
thf(fact_309_equalityD1,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% equalityD1
thf(fact_310_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_311_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_312_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A3: set_complex,B2: set_complex] :
! [X3: complex] :
( ( member_complex @ X3 @ A3 )
=> ( member_complex @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_313_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B2: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A3 )
=> ( member_list_a @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_314_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B2: set_o] :
! [X3: $o] :
( ( member_o @ X3 @ A3 )
=> ( member_o @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_315_equalityE,axiom,
! [A: set_a,B: set_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ B @ A ) ) ) ).
% equalityE
thf(fact_316_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_317_equalityE,axiom,
! [A: set_complex,B: set_complex] :
( ( A = B )
=> ~ ( ( ord_le211207098394363844omplex @ A @ B )
=> ~ ( ord_le211207098394363844omplex @ B @ A ) ) ) ).
% equalityE
thf(fact_318_equalityE,axiom,
! [A: set_list_a,B: set_list_a] :
( ( A = B )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B )
=> ~ ( ord_le8861187494160871172list_a @ B @ A ) ) ) ).
% equalityE
thf(fact_319_equalityE,axiom,
! [A: set_o,B: set_o] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_o @ A @ B )
=> ~ ( ord_less_eq_set_o @ B @ A ) ) ) ).
% equalityE
thf(fact_320_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_321_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_322_subsetD,axiom,
! [A: set_complex,B: set_complex,C: complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( member_complex @ C @ A )
=> ( member_complex @ C @ B ) ) ) ).
% subsetD
thf(fact_323_subsetD,axiom,
! [A: set_list_a,B: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B ) ) ) ).
% subsetD
thf(fact_324_subsetD,axiom,
! [A: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_325_in__mono,axiom,
! [A: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_326_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_327_in__mono,axiom,
! [A: set_complex,B: set_complex,X2: complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( member_complex @ X2 @ A )
=> ( member_complex @ X2 @ B ) ) ) ).
% in_mono
thf(fact_328_in__mono,axiom,
! [A: set_list_a,B: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( member_list_a @ X2 @ A )
=> ( member_list_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_329_in__mono,axiom,
! [A: set_o,B: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X2 @ A )
=> ( member_o @ X2 @ B ) ) ) ).
% in_mono
thf(fact_330_neq__if__length__neq,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs2 )
!= ( size_size_list_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_331_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_332_neq__if__length__neq,axiom,
! [Xs2: list_complex,Ys: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs2 )
!= ( size_s3451745648224563538omplex @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_333_neq__if__length__neq,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
!= ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_334_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_335_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_336_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_complex] :
( ( size_s3451745648224563538omplex @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_337_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_a] :
( ( size_s349497388124573686list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_338_finite__Collect__subsets,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B2: set_a] : ( ord_less_eq_set_a @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_339_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_340_finite__Collect__subsets,axiom,
! [A: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ( finite6551019134538273531omplex
@ ( collect_set_complex
@ ^ [B2: set_complex] : ( ord_le211207098394363844omplex @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_341_finite__Collect__subsets,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [B2: set_list_a] : ( ord_le8861187494160871172list_a @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_342_finite__Collect__subsets,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( finite_finite_set_o
@ ( collect_set_o
@ ^ [B2: set_o] : ( ord_less_eq_set_o @ B2 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_343_finite__Collect__disjI,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
& ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_344_finite__Collect__disjI,axiom,
! [P: list_complex > $o,Q: list_complex > $o] :
( ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [X3: list_complex] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite8712137658972009173omplex @ ( collect_list_complex @ P ) )
& ( finite8712137658972009173omplex @ ( collect_list_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_345_finite__Collect__disjI,axiom,
! [P: list_list_a > $o,Q: list_list_a > $o] :
( ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P ) )
& ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_346_finite__Collect__disjI,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( finite_finite_o
@ ( collect_o
@ ^ [X3: $o] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_o @ ( collect_o @ P ) )
& ( finite_finite_o @ ( collect_o @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_347_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_348_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_349_finite__Collect__disjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X3: complex] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
& ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_350_finite__Collect__disjI,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( P @ X3 )
| ( Q @ X3 ) ) ) )
= ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
& ( finite_finite_list_a @ ( collect_list_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_351_finite__Collect__conjI,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
| ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X3: list_nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_352_finite__Collect__conjI,axiom,
! [P: list_complex > $o,Q: list_complex > $o] :
( ( ( finite8712137658972009173omplex @ ( collect_list_complex @ P ) )
| ( finite8712137658972009173omplex @ ( collect_list_complex @ Q ) ) )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [X3: list_complex] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_353_finite__Collect__conjI,axiom,
! [P: list_list_a > $o,Q: list_list_a > $o] :
( ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P ) )
| ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q ) ) )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [X3: list_list_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_354_finite__Collect__conjI,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( ( finite_finite_o @ ( collect_o @ P ) )
| ( finite_finite_o @ ( collect_o @ Q ) ) )
=> ( finite_finite_o
@ ( collect_o
@ ^ [X3: $o] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_355_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X3: a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_356_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_357_finite__Collect__conjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
| ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X3: complex] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_358_finite__Collect__conjI,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
| ( finite_finite_list_a @ ( collect_list_a @ Q ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X3: list_a] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_359_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_list_nat,C2: nat] :
( ! [G: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ G @ F2 )
=> ( ( finite8100373058378681591st_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ G ) @ C2 ) ) )
=> ( ( finite8100373058378681591st_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_list_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_360_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_list_complex,C2: nat] :
( ! [G: set_list_complex] :
( ( ord_le3922870914418331732omplex @ G @ F2 )
=> ( ( finite8712137658972009173omplex @ G )
=> ( ord_less_eq_nat @ ( finite5120063068150530198omplex @ G ) @ C2 ) ) )
=> ( ( finite8712137658972009173omplex @ F2 )
& ( ord_less_eq_nat @ ( finite5120063068150530198omplex @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_361_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_list_list_a,C2: nat] :
( ! [G: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ G @ F2 )
=> ( ( finite1660835950917165235list_a @ G )
=> ( ord_less_eq_nat @ ( finite9134805042761151410list_a @ G ) @ C2 ) ) )
=> ( ( finite1660835950917165235list_a @ F2 )
& ( ord_less_eq_nat @ ( finite9134805042761151410list_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_362_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C2: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F2 )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_363_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_364_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_complex,C2: nat] :
( ! [G: set_complex] :
( ( ord_le211207098394363844omplex @ G @ F2 )
=> ( ( finite3207457112153483333omplex @ G )
=> ( ord_less_eq_nat @ ( finite_card_complex @ G ) @ C2 ) ) )
=> ( ( finite3207457112153483333omplex @ F2 )
& ( ord_less_eq_nat @ ( finite_card_complex @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_365_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_list_a,C2: nat] :
( ! [G: set_list_a] :
( ( ord_le8861187494160871172list_a @ G @ F2 )
=> ( ( finite_finite_list_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_list_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_list_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_366_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_o,C2: nat] :
( ! [G: set_o] :
( ( ord_less_eq_set_o @ G @ F2 )
=> ( ( finite_finite_o @ G )
=> ( ord_less_eq_nat @ ( finite_card_o @ G ) @ C2 ) ) )
=> ( ( finite_finite_o @ F2 )
& ( ord_less_eq_nat @ ( finite_card_o @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_367_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_list_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ S ) )
=> ~ ! [T2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ T2 @ S )
=> ( ( ( finite_card_list_nat @ T2 )
= N )
=> ~ ( finite8100373058378681591st_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_368_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_list_complex] :
( ( ord_less_eq_nat @ N @ ( finite5120063068150530198omplex @ S ) )
=> ~ ! [T2: set_list_complex] :
( ( ord_le3922870914418331732omplex @ T2 @ S )
=> ( ( ( finite5120063068150530198omplex @ T2 )
= N )
=> ~ ( finite8712137658972009173omplex @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_369_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_list_list_a] :
( ( ord_less_eq_nat @ N @ ( finite9134805042761151410list_a @ S ) )
=> ~ ! [T2: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ T2 @ S )
=> ( ( ( finite9134805042761151410list_a @ T2 )
= N )
=> ~ ( finite1660835950917165235list_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_370_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T2: set_a] :
( ( ord_less_eq_set_a @ T2 @ S )
=> ( ( ( finite_card_a @ T2 )
= N )
=> ~ ( finite_finite_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_371_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T2: set_nat] :
( ( ord_less_eq_set_nat @ T2 @ S )
=> ( ( ( finite_card_nat @ T2 )
= N )
=> ~ ( finite_finite_nat @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_372_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_complex] :
( ( ord_less_eq_nat @ N @ ( finite_card_complex @ S ) )
=> ~ ! [T2: set_complex] :
( ( ord_le211207098394363844omplex @ T2 @ S )
=> ( ( ( finite_card_complex @ T2 )
= N )
=> ~ ( finite3207457112153483333omplex @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_373_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_list_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ S ) )
=> ~ ! [T2: set_list_a] :
( ( ord_le8861187494160871172list_a @ T2 @ S )
=> ( ( ( finite_card_list_a @ T2 )
= N )
=> ~ ( finite_finite_list_a @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_374_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_o] :
( ( ord_less_eq_nat @ N @ ( finite_card_o @ S ) )
=> ~ ! [T2: set_o] :
( ( ord_less_eq_set_o @ T2 @ S )
=> ( ( ( finite_card_o @ T2 )
= N )
=> ~ ( finite_finite_o @ T2 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_375_exists__subset__between,axiom,
! [A: set_list_nat,N: nat,C2: set_list_nat] :
( ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ C2 ) )
=> ( ( ord_le6045566169113846134st_nat @ A @ C2 )
=> ( ( finite8100373058378681591st_nat @ C2 )
=> ? [B6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B6 )
& ( ord_le6045566169113846134st_nat @ B6 @ C2 )
& ( ( finite_card_list_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_376_exists__subset__between,axiom,
! [A: set_list_complex,N: nat,C2: set_list_complex] :
( ( ord_less_eq_nat @ ( finite5120063068150530198omplex @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite5120063068150530198omplex @ C2 ) )
=> ( ( ord_le3922870914418331732omplex @ A @ C2 )
=> ( ( finite8712137658972009173omplex @ C2 )
=> ? [B6: set_list_complex] :
( ( ord_le3922870914418331732omplex @ A @ B6 )
& ( ord_le3922870914418331732omplex @ B6 @ C2 )
& ( ( finite5120063068150530198omplex @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_377_exists__subset__between,axiom,
! [A: set_list_list_a,N: nat,C2: set_list_list_a] :
( ( ord_less_eq_nat @ ( finite9134805042761151410list_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite9134805042761151410list_a @ C2 ) )
=> ( ( ord_le8488217952732425610list_a @ A @ C2 )
=> ( ( finite1660835950917165235list_a @ C2 )
=> ? [B6: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A @ B6 )
& ( ord_le8488217952732425610list_a @ B6 @ C2 )
& ( ( finite9134805042761151410list_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_378_exists__subset__between,axiom,
! [A: set_a,N: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B6: set_a] :
( ( ord_less_eq_set_a @ A @ B6 )
& ( ord_less_eq_set_a @ B6 @ C2 )
& ( ( finite_card_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_379_exists__subset__between,axiom,
! [A: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B6: set_nat] :
( ( ord_less_eq_set_nat @ A @ B6 )
& ( ord_less_eq_set_nat @ B6 @ C2 )
& ( ( finite_card_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_380_exists__subset__between,axiom,
! [A: set_complex,N: nat,C2: set_complex] :
( ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_complex @ C2 ) )
=> ( ( ord_le211207098394363844omplex @ A @ C2 )
=> ( ( finite3207457112153483333omplex @ C2 )
=> ? [B6: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B6 )
& ( ord_le211207098394363844omplex @ B6 @ C2 )
& ( ( finite_card_complex @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_381_exists__subset__between,axiom,
! [A: set_list_a,N: nat,C2: set_list_a] :
( ( ord_less_eq_nat @ ( finite_card_list_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ C2 ) )
=> ( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ( finite_finite_list_a @ C2 )
=> ? [B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B6 )
& ( ord_le8861187494160871172list_a @ B6 @ C2 )
& ( ( finite_card_list_a @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_382_exists__subset__between,axiom,
! [A: set_o,N: nat,C2: set_o] :
( ( ord_less_eq_nat @ ( finite_card_o @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_o @ C2 ) )
=> ( ( ord_less_eq_set_o @ A @ C2 )
=> ( ( finite_finite_o @ C2 )
=> ? [B6: set_o] :
( ( ord_less_eq_set_o @ A @ B6 )
& ( ord_less_eq_set_o @ B6 @ C2 )
& ( ( finite_card_o @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_383_card__seteq,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B ) @ ( finite_card_list_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_384_card__seteq,axiom,
! [B: set_list_complex,A: set_list_complex] :
( ( finite8712137658972009173omplex @ B )
=> ( ( ord_le3922870914418331732omplex @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite5120063068150530198omplex @ B ) @ ( finite5120063068150530198omplex @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_385_card__seteq,axiom,
! [B: set_list_list_a,A: set_list_list_a] :
( ( finite1660835950917165235list_a @ B )
=> ( ( ord_le8488217952732425610list_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite9134805042761151410list_a @ B ) @ ( finite9134805042761151410list_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_386_card__seteq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_387_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_388_card__seteq,axiom,
! [B: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B )
=> ( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_complex @ B ) @ ( finite_card_complex @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_389_card__seteq,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_list_a @ B ) @ ( finite_card_list_a @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_390_card__seteq,axiom,
! [B: set_o,A: set_o] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_o @ B ) @ ( finite_card_o @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_391_card__mono,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B ) ) ) ) ).
% card_mono
thf(fact_392_card__mono,axiom,
! [B: set_list_complex,A: set_list_complex] :
( ( finite8712137658972009173omplex @ B )
=> ( ( ord_le3922870914418331732omplex @ A @ B )
=> ( ord_less_eq_nat @ ( finite5120063068150530198omplex @ A ) @ ( finite5120063068150530198omplex @ B ) ) ) ) ).
% card_mono
thf(fact_393_card__mono,axiom,
! [B: set_list_list_a,A: set_list_list_a] :
( ( finite1660835950917165235list_a @ B )
=> ( ( ord_le8488217952732425610list_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite9134805042761151410list_a @ A ) @ ( finite9134805042761151410list_a @ B ) ) ) ) ).
% card_mono
thf(fact_394_card__mono,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ).
% card_mono
thf(fact_395_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_396_card__mono,axiom,
! [B: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B )
=> ( ( ord_le211207098394363844omplex @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_complex @ B ) ) ) ) ).
% card_mono
thf(fact_397_card__mono,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ A ) @ ( finite_card_list_a @ B ) ) ) ) ).
% card_mono
thf(fact_398_card__mono,axiom,
! [B: set_o,A: set_o] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_o @ A ) @ ( finite_card_o @ B ) ) ) ) ).
% card_mono
thf(fact_399_infinite__arbitrarily__large,axiom,
! [A: set_list_nat,N: nat] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ? [B6: set_list_nat] :
( ( finite8100373058378681591st_nat @ B6 )
& ( ( finite_card_list_nat @ B6 )
= N )
& ( ord_le6045566169113846134st_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_400_infinite__arbitrarily__large,axiom,
! [A: set_list_complex,N: nat] :
( ~ ( finite8712137658972009173omplex @ A )
=> ? [B6: set_list_complex] :
( ( finite8712137658972009173omplex @ B6 )
& ( ( finite5120063068150530198omplex @ B6 )
= N )
& ( ord_le3922870914418331732omplex @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_401_infinite__arbitrarily__large,axiom,
! [A: set_list_list_a,N: nat] :
( ~ ( finite1660835950917165235list_a @ A )
=> ? [B6: set_list_list_a] :
( ( finite1660835950917165235list_a @ B6 )
& ( ( finite9134805042761151410list_a @ B6 )
= N )
& ( ord_le8488217952732425610list_a @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_402_infinite__arbitrarily__large,axiom,
! [A: set_a,N: nat] :
( ~ ( finite_finite_a @ A )
=> ? [B6: set_a] :
( ( finite_finite_a @ B6 )
& ( ( finite_card_a @ B6 )
= N )
& ( ord_less_eq_set_a @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_403_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N )
& ( ord_less_eq_set_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_404_infinite__arbitrarily__large,axiom,
! [A: set_complex,N: nat] :
( ~ ( finite3207457112153483333omplex @ A )
=> ? [B6: set_complex] :
( ( finite3207457112153483333omplex @ B6 )
& ( ( finite_card_complex @ B6 )
= N )
& ( ord_le211207098394363844omplex @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_405_infinite__arbitrarily__large,axiom,
! [A: set_list_a,N: nat] :
( ~ ( finite_finite_list_a @ A )
=> ? [B6: set_list_a] :
( ( finite_finite_list_a @ B6 )
& ( ( finite_card_list_a @ B6 )
= N )
& ( ord_le8861187494160871172list_a @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_406_infinite__arbitrarily__large,axiom,
! [A: set_o,N: nat] :
( ~ ( finite_finite_o @ A )
=> ? [B6: set_o] :
( ( finite_finite_o @ B6 )
& ( ( finite_card_o @ B6 )
= N )
& ( ord_less_eq_set_o @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_407_card__subset__eq,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( finite_card_list_nat @ A )
= ( finite_card_list_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_408_card__subset__eq,axiom,
! [B: set_list_complex,A: set_list_complex] :
( ( finite8712137658972009173omplex @ B )
=> ( ( ord_le3922870914418331732omplex @ A @ B )
=> ( ( ( finite5120063068150530198omplex @ A )
= ( finite5120063068150530198omplex @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_409_card__subset__eq,axiom,
! [B: set_list_list_a,A: set_list_list_a] :
( ( finite1660835950917165235list_a @ B )
=> ( ( ord_le8488217952732425610list_a @ A @ B )
=> ( ( ( finite9134805042761151410list_a @ A )
= ( finite9134805042761151410list_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_410_card__subset__eq,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_411_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_412_card__subset__eq,axiom,
! [B: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B )
=> ( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( ( finite_card_complex @ A )
= ( finite_card_complex @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_413_card__subset__eq,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( ( finite_card_list_a @ A )
= ( finite_card_list_a @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_414_card__subset__eq,axiom,
! [B: set_o,A: set_o] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ( ( finite_card_o @ A )
= ( finite_card_o @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_415_card__le__if__inj__on__rel,axiom,
! [B: set_o,A: set_o,R: $o > $o > $o] :
( ( finite_finite_o @ B )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ? [B7: $o] :
( ( member_o @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: $o,A22: $o,B5: $o] :
( ( member_o @ A1 @ A )
=> ( ( member_o @ A22 @ A )
=> ( ( member_o @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_o @ A ) @ ( finite_card_o @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_416_card__le__if__inj__on__rel,axiom,
! [B: set_o,A: set_a,R: a > $o > $o] :
( ( finite_finite_o @ B )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ? [B7: $o] :
( ( member_o @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: a,A22: a,B5: $o] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_o @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_o @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_417_card__le__if__inj__on__rel,axiom,
! [B: set_o,A: set_nat,R: nat > $o > $o] :
( ( finite_finite_o @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B7: $o] :
( ( member_o @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B5: $o] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_o @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_o @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_418_card__le__if__inj__on__rel,axiom,
! [B: set_o,A: set_complex,R: complex > $o > $o] :
( ( finite_finite_o @ B )
=> ( ! [A5: complex] :
( ( member_complex @ A5 @ A )
=> ? [B7: $o] :
( ( member_o @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: complex,A22: complex,B5: $o] :
( ( member_complex @ A1 @ A )
=> ( ( member_complex @ A22 @ A )
=> ( ( member_o @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_o @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_419_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_o,R: $o > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ? [B7: a] :
( ( member_a @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: $o,A22: $o,B5: a] :
( ( member_o @ A1 @ A )
=> ( ( member_o @ A22 @ A )
=> ( ( member_a @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_o @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_420_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_a,R: a > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ? [B7: a] :
( ( member_a @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: a,A22: a,B5: a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_a @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_421_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B7: a] :
( ( member_a @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B5: a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_a @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_422_card__le__if__inj__on__rel,axiom,
! [B: set_a,A: set_complex,R: complex > a > $o] :
( ( finite_finite_a @ B )
=> ( ! [A5: complex] :
( ( member_complex @ A5 @ A )
=> ? [B7: a] :
( ( member_a @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: complex,A22: complex,B5: a] :
( ( member_complex @ A1 @ A )
=> ( ( member_complex @ A22 @ A )
=> ( ( member_a @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_423_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_o,R: $o > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ? [B7: nat] :
( ( member_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: $o,A22: $o,B5: nat] :
( ( member_o @ A1 @ A )
=> ( ( member_o @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_o @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_424_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ? [B7: nat] :
( ( member_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: a,A22: a,B5: nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_425_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_426_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_427_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_428_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_429_not__finite__existsD,axiom,
! [P: list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
=> ? [X_1: list_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_430_not__finite__existsD,axiom,
! [P: list_complex > $o] :
( ~ ( finite8712137658972009173omplex @ ( collect_list_complex @ P ) )
=> ? [X_1: list_complex] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_431_not__finite__existsD,axiom,
! [P: list_list_a > $o] :
( ~ ( finite1660835950917165235list_a @ ( collect_list_list_a @ P ) )
=> ? [X_1: list_list_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_432_not__finite__existsD,axiom,
! [P: $o > $o] :
( ~ ( finite_finite_o @ ( collect_o @ P ) )
=> ? [X_1: $o] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_433_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_434_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_435_not__finite__existsD,axiom,
! [P: complex > $o] :
( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ? [X_1: complex] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_436_not__finite__existsD,axiom,
! [P: list_a > $o] :
( ~ ( finite_finite_list_a @ ( collect_list_a @ P ) )
=> ? [X_1: list_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_437_pigeonhole__infinite__rel,axiom,
! [A: set_o,B: set_a,R2: $o > a > $o] :
( ~ ( finite_finite_o @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R2 @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_438_pigeonhole__infinite__rel,axiom,
! [A: set_o,B: set_nat,R2: $o > nat > $o] :
( ~ ( finite_finite_o @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_439_pigeonhole__infinite__rel,axiom,
! [A: set_o,B: set_complex,R2: $o > complex > $o] :
( ~ ( finite_finite_o @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B )
& ( R2 @ X @ Xa ) ) )
=> ? [X: complex] :
( ( member_complex @ X @ B )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_440_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_a,R2: a > a > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R2 @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_441_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_nat,R2: a > nat > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_442_pigeonhole__infinite__rel,axiom,
! [A: set_a,B: set_complex,R2: a > complex > $o] :
( ~ ( finite_finite_a @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B )
& ( R2 @ X @ Xa ) ) )
=> ? [X: complex] :
( ( member_complex @ X @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_443_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_a,R2: nat > a > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R2 @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_444_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_nat,R2: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R2 @ X @ Xa ) ) )
=> ? [X: nat] :
( ( member_nat @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_445_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B: set_complex,R2: nat > complex > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ? [Xa: complex] :
( ( member_complex @ Xa @ B )
& ( R2 @ X @ Xa ) ) )
=> ? [X: complex] :
( ( member_complex @ X @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_446_pigeonhole__infinite__rel,axiom,
! [A: set_complex,B: set_a,R2: complex > a > $o] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_a @ B )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R2 @ X @ Xa ) ) )
=> ? [X: a] :
( ( member_a @ X @ B )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A4: complex] :
( ( member_complex @ A4 @ A )
& ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_447_finite__has__maximal2,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ( ord_less_eq_o @ A2 @ X )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_448_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ A2 @ X )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_449_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ A2 @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_450_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ A2 @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_451_finite__has__maximal2,axiom,
! [A: set_set_complex,A2: set_complex] :
( ( finite6551019134538273531omplex @ A )
=> ( ( member_set_complex @ A2 @ A )
=> ? [X: set_complex] :
( ( member_set_complex @ X @ A )
& ( ord_le211207098394363844omplex @ A2 @ X )
& ! [Xa: set_complex] :
( ( member_set_complex @ Xa @ A )
=> ( ( ord_le211207098394363844omplex @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_452_finite__has__maximal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X: set_list_a] :
( ( member_set_list_a @ X @ A )
& ( ord_le8861187494160871172list_a @ A2 @ X )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_453_finite__has__maximal2,axiom,
! [A: set_set_o,A2: set_o] :
( ( finite_finite_set_o @ A )
=> ( ( member_set_o @ A2 @ A )
=> ? [X: set_o] :
( ( member_set_o @ X @ A )
& ( ord_less_eq_set_o @ A2 @ X )
& ! [Xa: set_o] :
( ( member_set_o @ Xa @ A )
=> ( ( ord_less_eq_set_o @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_454_finite__has__minimal2,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ( ord_less_eq_o @ X @ A2 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_455_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X: set_a] :
( ( member_set_a @ X @ A )
& ( ord_less_eq_set_a @ X @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_456_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_457_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_458_finite__has__minimal2,axiom,
! [A: set_set_complex,A2: set_complex] :
( ( finite6551019134538273531omplex @ A )
=> ( ( member_set_complex @ A2 @ A )
=> ? [X: set_complex] :
( ( member_set_complex @ X @ A )
& ( ord_le211207098394363844omplex @ X @ A2 )
& ! [Xa: set_complex] :
( ( member_set_complex @ Xa @ A )
=> ( ( ord_le211207098394363844omplex @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_459_finite__has__minimal2,axiom,
! [A: set_set_list_a,A2: set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ( member_set_list_a @ A2 @ A )
=> ? [X: set_list_a] :
( ( member_set_list_a @ X @ A )
& ( ord_le8861187494160871172list_a @ X @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_460_finite__has__minimal2,axiom,
! [A: set_set_o,A2: set_o] :
( ( finite_finite_set_o @ A )
=> ( ( member_set_o @ A2 @ A )
=> ? [X: set_o] :
( ( member_set_o @ X @ A )
& ( ord_less_eq_set_o @ X @ A2 )
& ! [Xa: set_o] :
( ( member_set_o @ Xa @ A )
=> ( ( ord_less_eq_set_o @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_461_finite__subset,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_462_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_463_finite__subset,axiom,
! [A: set_complex,B: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ( finite3207457112153483333omplex @ B )
=> ( finite3207457112153483333omplex @ A ) ) ) ).
% finite_subset
thf(fact_464_finite__subset,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ( finite_finite_list_a @ B )
=> ( finite_finite_list_a @ A ) ) ) ).
% finite_subset
thf(fact_465_finite__subset,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( finite_finite_o @ B )
=> ( finite_finite_o @ A ) ) ) ).
% finite_subset
thf(fact_466_infinite__super,axiom,
! [S: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S @ T3 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T3 ) ) ) ).
% infinite_super
thf(fact_467_infinite__super,axiom,
! [S: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S @ T3 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_468_infinite__super,axiom,
! [S: set_complex,T3: set_complex] :
( ( ord_le211207098394363844omplex @ S @ T3 )
=> ( ~ ( finite3207457112153483333omplex @ S )
=> ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% infinite_super
thf(fact_469_infinite__super,axiom,
! [S: set_list_a,T3: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T3 )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T3 ) ) ) ).
% infinite_super
thf(fact_470_infinite__super,axiom,
! [S: set_o,T3: set_o] :
( ( ord_less_eq_set_o @ S @ T3 )
=> ( ~ ( finite_finite_o @ S )
=> ~ ( finite_finite_o @ T3 ) ) ) ).
% infinite_super
thf(fact_471_rev__finite__subset,axiom,
! [B: set_a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_472_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_473_rev__finite__subset,axiom,
! [B: set_complex,A: set_complex] :
( ( finite3207457112153483333omplex @ B )
=> ( ( ord_le211207098394363844omplex @ A @ B )
=> ( finite3207457112153483333omplex @ A ) ) ) ).
% rev_finite_subset
thf(fact_474_rev__finite__subset,axiom,
! [B: set_list_a,A: set_list_a] :
( ( finite_finite_list_a @ B )
=> ( ( ord_le8861187494160871172list_a @ A @ B )
=> ( finite_finite_list_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_475_rev__finite__subset,axiom,
! [B: set_o,A: set_o] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( finite_finite_o @ A ) ) ) ).
% rev_finite_subset
thf(fact_476_pred__subset__eq,axiom,
! [R2: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R2 )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ord_less_eq_set_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_477_pred__subset__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X3: nat] : ( member_nat @ X3 @ R2 )
@ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
= ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_478_pred__subset__eq,axiom,
! [R2: set_complex,S: set_complex] :
( ( ord_le4573692005234683329plex_o
@ ^ [X3: complex] : ( member_complex @ X3 @ R2 )
@ ^ [X3: complex] : ( member_complex @ X3 @ S ) )
= ( ord_le211207098394363844omplex @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_479_pred__subset__eq,axiom,
! [R2: set_list_a,S: set_list_a] :
( ( ord_less_eq_list_a_o
@ ^ [X3: list_a] : ( member_list_a @ X3 @ R2 )
@ ^ [X3: list_a] : ( member_list_a @ X3 @ S ) )
= ( ord_le8861187494160871172list_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_480_pred__subset__eq,axiom,
! [R2: set_o,S: set_o] :
( ( ord_less_eq_o_o
@ ^ [X3: $o] : ( member_o @ X3 @ R2 )
@ ^ [X3: $o] : ( member_o @ X3 @ S ) )
= ( ord_less_eq_set_o @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_481_set__n__lists,axiom,
! [N: nat,Xs2: list_a] :
( ( set_list_a2 @ ( n_lists_a @ N @ Xs2 ) )
= ( collect_list_a
@ ^ [Ys2: list_a] :
( ( ( size_size_list_a @ Ys2 )
= N )
& ( ord_less_eq_set_a @ ( set_a2 @ Ys2 ) @ ( set_a2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_482_set__n__lists,axiom,
! [N: nat,Xs2: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_483_set__n__lists,axiom,
! [N: nat,Xs2: list_complex] :
( ( set_list_complex2 @ ( n_lists_complex @ N @ Xs2 ) )
= ( collect_list_complex
@ ^ [Ys2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Ys2 )
= N )
& ( ord_le211207098394363844omplex @ ( set_complex2 @ Ys2 ) @ ( set_complex2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_484_set__n__lists,axiom,
! [N: nat,Xs2: list_list_a] :
( ( set_list_list_a2 @ ( n_lists_list_a @ N @ Xs2 ) )
= ( collect_list_list_a
@ ^ [Ys2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Ys2 )
= N )
& ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ys2 ) @ ( set_list_a2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_485_set__n__lists,axiom,
! [N: nat,Xs2: list_o] :
( ( set_list_o2 @ ( n_lists_o @ N @ Xs2 ) )
= ( collect_list_o
@ ^ [Ys2: list_o] :
( ( ( size_size_list_o @ Ys2 )
= N )
& ( ord_less_eq_set_o @ ( set_o2 @ Ys2 ) @ ( set_o2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_486_finite__lists__distinct__length__eq,axiom,
! [A: set_a,N: nat] :
( ( finite_finite_a @ A )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= N )
& ( distinct_a @ Xs )
& ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_487_finite__lists__distinct__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= N )
& ( distinct_nat @ Xs )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_488_finite__lists__distinct__length__eq,axiom,
! [A: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A )
=> ( finite8712137658972009173omplex
@ ( collect_list_complex
@ ^ [Xs: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs )
= N )
& ( distinct_complex @ Xs )
& ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_489_finite__lists__distinct__length__eq,axiom,
! [A: set_list_a,N: nat] :
( ( finite_finite_list_a @ A )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [Xs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= N )
& ( distinct_list_a @ Xs )
& ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_490_finite__lists__distinct__length__eq,axiom,
! [A: set_o,N: nat] :
( ( finite_finite_o @ A )
=> ( finite_finite_list_o
@ ( collect_list_o
@ ^ [Xs: list_o] :
( ( ( size_size_list_o @ Xs )
= N )
& ( distinct_o @ Xs )
& ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_491_Fpow__def,axiom,
( finite_Fpow_a
= ( ^ [A3: set_a] :
( collect_set_a
@ ^ [X6: set_a] :
( ( ord_less_eq_set_a @ X6 @ A3 )
& ( finite_finite_a @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_492_Fpow__def,axiom,
( finite_Fpow_nat
= ( ^ [A3: set_nat] :
( collect_set_nat
@ ^ [X6: set_nat] :
( ( ord_less_eq_set_nat @ X6 @ A3 )
& ( finite_finite_nat @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_493_Fpow__def,axiom,
( finite_Fpow_complex
= ( ^ [A3: set_complex] :
( collect_set_complex
@ ^ [X6: set_complex] :
( ( ord_le211207098394363844omplex @ X6 @ A3 )
& ( finite3207457112153483333omplex @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_494_Fpow__def,axiom,
( finite_Fpow_list_a
= ( ^ [A3: set_list_a] :
( collect_set_list_a
@ ^ [X6: set_list_a] :
( ( ord_le8861187494160871172list_a @ X6 @ A3 )
& ( finite_finite_list_a @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_495_Fpow__def,axiom,
( finite_Fpow_o
= ( ^ [A3: set_o] :
( collect_set_o
@ ^ [X6: set_o] :
( ( ord_less_eq_set_o @ X6 @ A3 )
& ( finite_finite_o @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_496_card__lists__length__le,axiom,
! [A: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( finite7325466520557071688st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_list_nat @ A ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_497_card__lists__length__le,axiom,
! [A: set_list_complex,N: nat] :
( ( finite8712137658972009173omplex @ A )
=> ( ( finite5336269520247027750omplex
@ ( collec1601192001008753443omplex
@ ^ [Xs: list_list_complex] :
( ( ord_le3922870914418331732omplex @ ( set_list_complex2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_s7907857696548412130omplex @ Xs ) @ N ) ) ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite5120063068150530198omplex @ A ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_498_card__lists__length__le,axiom,
! [A: set_list_list_a,N: nat] :
( ( finite1660835950917165235list_a @ A )
=> ( ( finite4595494376813527864list_a
@ ( collec1292721268053437947list_a
@ ^ [Xs: list_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_s2403821588304063868list_a @ Xs ) @ N ) ) ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite9134805042761151410list_a @ A ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_499_card__lists__length__le,axiom,
! [A: set_a,N: nat] :
( ( finite_finite_a @ A )
=> ( ( finite_card_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_a @ A ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_500_card__lists__length__le,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_nat @ A ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_501_card__lists__length__le,axiom,
! [A: set_complex,N: nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite5120063068150530198omplex
@ ( collect_list_complex
@ ^ [Xs: list_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_complex @ A ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_502_card__lists__length__le,axiom,
! [A: set_list_a,N: nat] :
( ( finite_finite_list_a @ A )
=> ( ( finite9134805042761151410list_a
@ ( collect_list_list_a
@ ^ [Xs: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N ) ) ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_list_a @ A ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_503_card__lists__length__le,axiom,
! [A: set_o,N: nat] :
( ( finite_finite_o @ A )
=> ( ( finite_card_list_o
@ ( collect_list_o
@ ^ [Xs: list_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) )
= ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_o @ A ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_504_surj__card__le,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_505_surj__card__le,axiom,
! [A: set_nat,B: set_a,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_506_surj__card__le,axiom,
! [A: set_complex,B: set_a,F: complex > a] :
( ( finite3207457112153483333omplex @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_complex_a @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ B ) @ ( finite_card_complex @ A ) ) ) ) ).
% surj_card_le
thf(fact_507_surj__card__le,axiom,
! [A: set_a,B: set_nat,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_508_surj__card__le,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_509_surj__card__le,axiom,
! [A: set_complex,B: set_nat,F: complex > nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_complex_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_complex @ A ) ) ) ) ).
% surj_card_le
thf(fact_510_surj__card__le,axiom,
! [A: set_a,B: set_complex,F: a > complex] :
( ( finite_finite_a @ A )
=> ( ( ord_le211207098394363844omplex @ B @ ( image_a_complex @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_complex @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_511_surj__card__le,axiom,
! [A: set_nat,B: set_complex,F: nat > complex] :
( ( finite_finite_nat @ A )
=> ( ( ord_le211207098394363844omplex @ B @ ( image_nat_complex @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_complex @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_512_surj__card__le,axiom,
! [A: set_complex,B: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( ord_le211207098394363844omplex @ B @ ( image_1468599708987790691omplex @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_complex @ B ) @ ( finite_card_complex @ A ) ) ) ) ).
% surj_card_le
thf(fact_513_surj__card__le,axiom,
! [A: set_a,B: set_o,F: a > $o] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_o @ B @ ( image_a_o @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_o @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_514_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A ) )
= ( finite_card_nat @ A ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_515_GreatestI2__order,axiom,
! [P: set_a > $o,X2: set_a,Q: set_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ! [X: set_a] :
( ( P @ X )
=> ( ! [Y5: set_a] :
( ( P @ Y5 )
=> ( ord_less_eq_set_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_516_GreatestI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ! [X: set_nat] :
( ( P @ X )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_517_GreatestI2__order,axiom,
! [P: set_complex > $o,X2: set_complex,Q: set_complex > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_complex] :
( ( P @ Y2 )
=> ( ord_le211207098394363844omplex @ Y2 @ X2 ) )
=> ( ! [X: set_complex] :
( ( P @ X )
=> ( ! [Y5: set_complex] :
( ( P @ Y5 )
=> ( ord_le211207098394363844omplex @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_95770167153410891omplex @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_518_GreatestI2__order,axiom,
! [P: set_list_a > $o,X2: set_list_a,Q: set_list_a > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_list_a] :
( ( P @ Y2 )
=> ( ord_le8861187494160871172list_a @ Y2 @ X2 ) )
=> ( ! [X: set_list_a] :
( ( P @ X )
=> ( ! [Y5: set_list_a] :
( ( P @ Y5 )
=> ( ord_le8861187494160871172list_a @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_733672244956367037list_a @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_519_GreatestI2__order,axiom,
! [P: set_o > $o,X2: set_o,Q: set_o > $o] :
( ( P @ X2 )
=> ( ! [Y2: set_o] :
( ( P @ Y2 )
=> ( ord_less_eq_set_o @ Y2 @ X2 ) )
=> ( ! [X: set_o] :
( ( P @ X )
=> ( ! [Y5: set_o] :
( ( P @ Y5 )
=> ( ord_less_eq_set_o @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_set_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_520_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_521_Greatest__equality,axiom,
! [P: set_a > $o,X2: set_a] :
( ( P @ X2 )
=> ( ! [Y2: set_a] :
( ( P @ Y2 )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ( order_Greatest_set_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_522_Greatest__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_523_Greatest__equality,axiom,
! [P: set_complex > $o,X2: set_complex] :
( ( P @ X2 )
=> ( ! [Y2: set_complex] :
( ( P @ Y2 )
=> ( ord_le211207098394363844omplex @ Y2 @ X2 ) )
=> ( ( order_95770167153410891omplex @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_524_Greatest__equality,axiom,
! [P: set_list_a > $o,X2: set_list_a] :
( ( P @ X2 )
=> ( ! [Y2: set_list_a] :
( ( P @ Y2 )
=> ( ord_le8861187494160871172list_a @ Y2 @ X2 ) )
=> ( ( order_733672244956367037list_a @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_525_Greatest__equality,axiom,
! [P: set_o > $o,X2: set_o] :
( ( P @ X2 )
=> ( ! [Y2: set_o] :
( ( P @ Y2 )
=> ( ord_less_eq_set_o @ Y2 @ X2 ) )
=> ( ( order_Greatest_set_o @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_526_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_527_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( set_o2 @ ( linord3142498349692569832_set_o @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_528_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_529_image__eqI,axiom,
! [B3: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_530_image__eqI,axiom,
! [B3: $o,F: nat > $o,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_o @ B3 @ ( image_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_531_image__eqI,axiom,
! [B3: a,F: nat > a,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_a @ B3 @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_532_image__eqI,axiom,
! [B3: complex,F: nat > complex,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_complex @ B3 @ ( image_nat_complex @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_533_image__eqI,axiom,
! [B3: nat,F: $o > nat,X2: $o,A: set_o] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_nat @ B3 @ ( image_o_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_534_image__eqI,axiom,
! [B3: $o,F: $o > $o,X2: $o,A: set_o] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_o @ B3 @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_535_image__eqI,axiom,
! [B3: a,F: $o > a,X2: $o,A: set_o] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_a @ B3 @ ( image_o_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_536_image__eqI,axiom,
! [B3: complex,F: $o > complex,X2: $o,A: set_o] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_o @ X2 @ A )
=> ( member_complex @ B3 @ ( image_o_complex @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_537_image__eqI,axiom,
! [B3: nat,F: a > nat,X2: a,A: set_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_nat @ B3 @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_538_image__eqI,axiom,
! [B3: $o,F: a > $o,X2: a,A: set_a] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_o @ B3 @ ( image_a_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_539_atMost__eq__iff,axiom,
! [X2: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X2 )
= ( set_ord_atMost_nat @ Y ) )
= ( X2 = Y ) ) ).
% atMost_eq_iff
thf(fact_540_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat
@ ^ [X3: nat] : X3
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_541_finite__imageI,axiom,
! [F2: set_a,H: a > a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_a @ ( image_a_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_542_finite__imageI,axiom,
! [F2: set_a,H: a > nat] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_nat @ ( image_a_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_543_finite__imageI,axiom,
! [F2: set_a,H: a > complex] :
( ( finite_finite_a @ F2 )
=> ( finite3207457112153483333omplex @ ( image_a_complex @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_544_finite__imageI,axiom,
! [F2: set_nat,H: nat > a] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_a @ ( image_nat_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_545_finite__imageI,axiom,
! [F2: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_546_finite__imageI,axiom,
! [F2: set_nat,H: nat > complex] :
( ( finite_finite_nat @ F2 )
=> ( finite3207457112153483333omplex @ ( image_nat_complex @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_547_finite__imageI,axiom,
! [F2: set_complex,H: complex > a] :
( ( finite3207457112153483333omplex @ F2 )
=> ( finite_finite_a @ ( image_complex_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_548_finite__imageI,axiom,
! [F2: set_complex,H: complex > nat] :
( ( finite3207457112153483333omplex @ F2 )
=> ( finite_finite_nat @ ( image_complex_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_549_finite__imageI,axiom,
! [F2: set_complex,H: complex > complex] :
( ( finite3207457112153483333omplex @ F2 )
=> ( finite3207457112153483333omplex @ ( image_1468599708987790691omplex @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_550_finite__imageI,axiom,
! [F2: set_a,H: a > list_a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_list_a @ ( image_a_list_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_551_atMost__iff,axiom,
! [I: $o,K: $o] :
( ( member_o @ I @ ( set_ord_atMost_o @ K ) )
= ( ord_less_eq_o @ I @ K ) ) ).
% atMost_iff
thf(fact_552_atMost__iff,axiom,
! [I: set_a,K: set_a] :
( ( member_set_a @ I @ ( set_ord_atMost_set_a @ K ) )
= ( ord_less_eq_set_a @ I @ K ) ) ).
% atMost_iff
thf(fact_553_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_554_atMost__iff,axiom,
! [I: set_complex,K: set_complex] :
( ( member_set_complex @ I @ ( set_or9043709113427266269omplex @ K ) )
= ( ord_le211207098394363844omplex @ I @ K ) ) ).
% atMost_iff
thf(fact_555_atMost__iff,axiom,
! [I: set_list_a,K: set_list_a] :
( ( member_set_list_a @ I @ ( set_or6279072120763780779list_a @ K ) )
= ( ord_le8861187494160871172list_a @ I @ K ) ) ).
% atMost_iff
thf(fact_556_atMost__iff,axiom,
! [I: set_o,K: set_o] :
( ( member_set_o @ I @ ( set_ord_atMost_set_o @ K ) )
= ( ord_less_eq_set_o @ I @ K ) ) ).
% atMost_iff
thf(fact_557_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_558_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_559_atMost__subset__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X2 ) @ ( set_ord_atMost_set_a @ Y ) )
= ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_560_atMost__subset__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X2 ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_561_atMost__subset__iff,axiom,
! [X2: set_complex,Y: set_complex] :
( ( ord_le4750530260501030778omplex @ ( set_or9043709113427266269omplex @ X2 ) @ ( set_or9043709113427266269omplex @ Y ) )
= ( ord_le211207098394363844omplex @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_562_atMost__subset__iff,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( ord_le8877086941679407844list_a @ ( set_or6279072120763780779list_a @ X2 ) @ ( set_or6279072120763780779list_a @ Y ) )
= ( ord_le8861187494160871172list_a @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_563_atMost__subset__iff,axiom,
! [X2: set_o,Y: set_o] :
( ( ord_le4374716579403074808_set_o @ ( set_ord_atMost_set_o @ X2 ) @ ( set_ord_atMost_set_o @ Y ) )
= ( ord_less_eq_set_o @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_564_atMost__subset__iff,axiom,
! [X2: $o,Y: $o] :
( ( ord_less_eq_set_o @ ( set_ord_atMost_o @ X2 ) @ ( set_ord_atMost_o @ Y ) )
= ( ord_less_eq_o @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_565_atMost__subset__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_566_image__Fpow__mono,axiom,
! [F: nat > set_nat,A: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_set_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_567_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_568_image__set__eqI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat,G2: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( member_nat @ ( G2 @ X ) @ A )
& ( ( F @ ( G2 @ X ) )
= X ) ) )
=> ( ( image_nat_nat @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_569_image__set__eqI,axiom,
! [A: set_nat,F: nat > $o,B: set_o,G2: $o > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( member_nat @ ( G2 @ X ) @ A )
& ( ( F @ ( G2 @ X ) )
= X ) ) )
=> ( ( image_nat_o @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_570_image__set__eqI,axiom,
! [A: set_nat,F: nat > a,B: set_a,G2: a > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( member_nat @ ( G2 @ X ) @ A )
& ( ( F @ ( G2 @ X ) )
= X ) ) )
=> ( ( image_nat_a @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_571_image__set__eqI,axiom,
! [A: set_nat,F: nat > complex,B: set_complex,G2: complex > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_complex @ ( F @ X ) @ B ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ B )
=> ( ( member_nat @ ( G2 @ X ) @ A )
& ( ( F @ ( G2 @ X ) )
= X ) ) )
=> ( ( image_nat_complex @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_572_image__set__eqI,axiom,
! [A: set_o,F: $o > nat,B: set_nat,G2: nat > $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( member_o @ ( G2 @ X ) @ A )
& ( ( F @ ( G2 @ X ) )
= X ) ) )
=> ( ( image_o_nat @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_573_image__set__eqI,axiom,
! [A: set_o,F: $o > $o,B: set_o,G2: $o > $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( member_o @ ( G2 @ X ) @ A )
& ( ( F @ ( G2 @ X ) )
= X ) ) )
=> ( ( image_o_o @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_574_image__set__eqI,axiom,
! [A: set_o,F: $o > a,B: set_a,G2: a > $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( member_o @ ( G2 @ X ) @ A )
& ( ( F @ ( G2 @ X ) )
= X ) ) )
=> ( ( image_o_a @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_575_image__set__eqI,axiom,
! [A: set_o,F: $o > complex,B: set_complex,G2: complex > $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_complex @ ( F @ X ) @ B ) )
=> ( ! [X: complex] :
( ( member_complex @ X @ B )
=> ( ( member_o @ ( G2 @ X ) @ A )
& ( ( F @ ( G2 @ X ) )
= X ) ) )
=> ( ( image_o_complex @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_576_image__set__eqI,axiom,
! [A: set_a,F: a > nat,B: set_nat,G2: nat > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( member_a @ ( G2 @ X ) @ A )
& ( ( F @ ( G2 @ X ) )
= X ) ) )
=> ( ( image_a_nat @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_577_image__set__eqI,axiom,
! [A: set_a,F: a > $o,B: set_o,G2: $o > a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( member_a @ ( G2 @ X ) @ A )
& ( ( F @ ( G2 @ X ) )
= X ) ) )
=> ( ( image_a_o @ F @ A )
= B ) ) ) ).
% image_set_eqI
thf(fact_578_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_579_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: $o,F: nat > $o] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_o @ B3 @ ( image_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_580_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: a,F: nat > a] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_a @ B3 @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_581_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: complex,F: nat > complex] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_complex @ B3 @ ( image_nat_complex @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_582_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B3: nat,F: $o > nat] :
( ( member_o @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat @ B3 @ ( image_o_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_583_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B3: $o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_o @ B3 @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_584_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B3: a,F: $o > a] :
( ( member_o @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_a @ B3 @ ( image_o_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_585_rev__image__eqI,axiom,
! [X2: $o,A: set_o,B3: complex,F: $o > complex] :
( ( member_o @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_complex @ B3 @ ( image_o_complex @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_586_rev__image__eqI,axiom,
! [X2: a,A: set_a,B3: nat,F: a > nat] :
( ( member_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat @ B3 @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_587_rev__image__eqI,axiom,
! [X2: a,A: set_a,B3: $o,F: a > $o] :
( ( member_a @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_o @ B3 @ ( image_a_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_588_ball__imageD,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ ( image_nat_set_nat @ F @ A ) )
=> ( P @ X ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_589_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
=> ( P @ X ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_590_image__cong,axiom,
! [M: set_nat,N5: set_nat,F: nat > set_nat,G2: nat > set_nat] :
( ( M = N5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ( F @ X )
= ( G2 @ X ) ) )
=> ( ( image_nat_set_nat @ F @ M )
= ( image_nat_set_nat @ G2 @ N5 ) ) ) ) ).
% image_cong
thf(fact_591_image__cong,axiom,
! [M: set_nat,N5: set_nat,F: nat > nat,G2: nat > nat] :
( ( M = N5 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ( F @ X )
= ( G2 @ X ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G2 @ N5 ) ) ) ) ).
% image_cong
thf(fact_592_bex__imageD,axiom,
! [F: nat > set_nat,A: set_nat,P: set_nat > $o] :
( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ ( image_nat_set_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_593_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_594_image__iff,axiom,
! [Z3: set_nat,F: nat > set_nat,A: set_nat] :
( ( member_set_nat @ Z3 @ ( image_nat_set_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_595_image__iff,axiom,
! [Z3: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_596_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_597_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_598_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_599_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > complex] :
( ( member_nat @ X2 @ A )
=> ( member_complex @ ( F @ X2 ) @ ( image_nat_complex @ F @ A ) ) ) ).
% imageI
thf(fact_600_imageI,axiom,
! [X2: $o,A: set_o,F: $o > nat] :
( ( member_o @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_o_nat @ F @ A ) ) ) ).
% imageI
thf(fact_601_imageI,axiom,
! [X2: $o,A: set_o,F: $o > $o] :
( ( member_o @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_602_imageI,axiom,
! [X2: $o,A: set_o,F: $o > a] :
( ( member_o @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_o_a @ F @ A ) ) ) ).
% imageI
thf(fact_603_imageI,axiom,
! [X2: $o,A: set_o,F: $o > complex] :
( ( member_o @ X2 @ A )
=> ( member_complex @ ( F @ X2 ) @ ( image_o_complex @ F @ A ) ) ) ).
% imageI
thf(fact_604_imageI,axiom,
! [X2: a,A: set_a,F: a > nat] :
( ( member_a @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_605_imageI,axiom,
! [X2: a,A: set_a,F: a > $o] :
( ( member_a @ X2 @ A )
=> ( member_o @ ( F @ X2 ) @ ( image_a_o @ F @ A ) ) ) ).
% imageI
thf(fact_606_imageE,axiom,
! [B3: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [X: nat] :
( ( B3
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_607_imageE,axiom,
! [B3: nat,F: $o > nat,A: set_o] :
( ( member_nat @ B3 @ ( image_o_nat @ F @ A ) )
=> ~ ! [X: $o] :
( ( B3
= ( F @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_608_imageE,axiom,
! [B3: nat,F: a > nat,A: set_a] :
( ( member_nat @ B3 @ ( image_a_nat @ F @ A ) )
=> ~ ! [X: a] :
( ( B3
= ( F @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_609_imageE,axiom,
! [B3: nat,F: complex > nat,A: set_complex] :
( ( member_nat @ B3 @ ( image_complex_nat @ F @ A ) )
=> ~ ! [X: complex] :
( ( B3
= ( F @ X ) )
=> ~ ( member_complex @ X @ A ) ) ) ).
% imageE
thf(fact_610_imageE,axiom,
! [B3: $o,F: nat > $o,A: set_nat] :
( ( member_o @ B3 @ ( image_nat_o @ F @ A ) )
=> ~ ! [X: nat] :
( ( B3
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_611_imageE,axiom,
! [B3: $o,F: $o > $o,A: set_o] :
( ( member_o @ B3 @ ( image_o_o @ F @ A ) )
=> ~ ! [X: $o] :
( ( B3
= ( F @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_612_imageE,axiom,
! [B3: $o,F: a > $o,A: set_a] :
( ( member_o @ B3 @ ( image_a_o @ F @ A ) )
=> ~ ! [X: a] :
( ( B3
= ( F @ X ) )
=> ~ ( member_a @ X @ A ) ) ) ).
% imageE
thf(fact_613_imageE,axiom,
! [B3: $o,F: complex > $o,A: set_complex] :
( ( member_o @ B3 @ ( image_complex_o @ F @ A ) )
=> ~ ! [X: complex] :
( ( B3
= ( F @ X ) )
=> ~ ( member_complex @ X @ A ) ) ) ).
% imageE
thf(fact_614_imageE,axiom,
! [B3: a,F: nat > a,A: set_nat] :
( ( member_a @ B3 @ ( image_nat_a @ F @ A ) )
=> ~ ! [X: nat] :
( ( B3
= ( F @ X ) )
=> ~ ( member_nat @ X @ A ) ) ) ).
% imageE
thf(fact_615_imageE,axiom,
! [B3: a,F: $o > a,A: set_o] :
( ( member_a @ B3 @ ( image_o_a @ F @ A ) )
=> ~ ! [X: $o] :
( ( B3
= ( F @ X ) )
=> ~ ( member_o @ X @ A ) ) ) ).
% imageE
thf(fact_616_image__image,axiom,
! [F: set_nat > set_nat,G2: nat > set_nat,A: set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( image_nat_set_nat @ G2 @ A ) )
= ( image_nat_set_nat
@ ^ [X3: nat] : ( F @ ( G2 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_617_image__image,axiom,
! [F: set_nat > nat,G2: nat > set_nat,A: set_nat] :
( ( image_set_nat_nat @ F @ ( image_nat_set_nat @ G2 @ A ) )
= ( image_nat_nat
@ ^ [X3: nat] : ( F @ ( G2 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_618_image__image,axiom,
! [F: nat > set_nat,G2: nat > nat,A: set_nat] :
( ( image_nat_set_nat @ F @ ( image_nat_nat @ G2 @ A ) )
= ( image_nat_set_nat
@ ^ [X3: nat] : ( F @ ( G2 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_619_image__image,axiom,
! [F: nat > nat,G2: nat > nat,A: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G2 @ A ) )
= ( image_nat_nat
@ ^ [X3: nat] : ( F @ ( G2 @ X3 ) )
@ A ) ) ).
% image_image
thf(fact_620_Compr__image__eq,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_621_Compr__image__eq,axiom,
! [F: complex > nat,A: set_complex,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_complex_nat @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_complex_nat @ F
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_622_Compr__image__eq,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_a_nat @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_623_Compr__image__eq,axiom,
! [F: $o > nat,A: set_o,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ ( image_o_nat @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_o_nat @ F
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_624_Compr__image__eq,axiom,
! [F: nat > complex,A: set_nat,P: complex > $o] :
( ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ ( image_nat_complex @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_nat_complex @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_625_Compr__image__eq,axiom,
! [F: complex > complex,A: set_complex,P: complex > $o] :
( ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ ( image_1468599708987790691omplex @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_1468599708987790691omplex @ F
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_626_Compr__image__eq,axiom,
! [F: a > complex,A: set_a,P: complex > $o] :
( ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ ( image_a_complex @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_a_complex @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_627_Compr__image__eq,axiom,
! [F: $o > complex,A: set_o,P: complex > $o] :
( ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ ( image_o_complex @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_o_complex @ F
@ ( collect_o
@ ^ [X3: $o] :
( ( member_o @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_628_Compr__image__eq,axiom,
! [F: nat > a,A: set_nat,P: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_nat_a @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_629_Compr__image__eq,axiom,
! [F: complex > a,A: set_complex,P: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_complex_a @ F @ A ) )
& ( P @ X3 ) ) )
= ( image_complex_a @ F
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_630_subset__image__iff,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_631_subset__image__iff,axiom,
! [B: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_632_subset__image__iff,axiom,
! [B: set_a,F: complex > a,A: set_complex] :
( ( ord_less_eq_set_a @ B @ ( image_complex_a @ F @ A ) )
= ( ? [AA: set_complex] :
( ( ord_le211207098394363844omplex @ AA @ A )
& ( B
= ( image_complex_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_633_subset__image__iff,axiom,
! [B: set_a,F: $o > a,A: set_o] :
( ( ord_less_eq_set_a @ B @ ( image_o_a @ F @ A ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A )
& ( B
= ( image_o_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_634_subset__image__iff,axiom,
! [B: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_635_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_636_subset__image__iff,axiom,
! [B: set_nat,F: complex > nat,A: set_complex] :
( ( ord_less_eq_set_nat @ B @ ( image_complex_nat @ F @ A ) )
= ( ? [AA: set_complex] :
( ( ord_le211207098394363844omplex @ AA @ A )
& ( B
= ( image_complex_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_637_subset__image__iff,axiom,
! [B: set_nat,F: $o > nat,A: set_o] :
( ( ord_less_eq_set_nat @ B @ ( image_o_nat @ F @ A ) )
= ( ? [AA: set_o] :
( ( ord_less_eq_set_o @ AA @ A )
& ( B
= ( image_o_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_638_subset__image__iff,axiom,
! [B: set_complex,F: a > complex,A: set_a] :
( ( ord_le211207098394363844omplex @ B @ ( image_a_complex @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_a_complex @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_639_subset__image__iff,axiom,
! [B: set_complex,F: nat > complex,A: set_nat] :
( ( ord_le211207098394363844omplex @ B @ ( image_nat_complex @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_complex @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_640_image__subset__iff,axiom,
! [F: nat > set_nat,A: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_set_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_641_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_642_subset__imageE,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_643_subset__imageE,axiom,
! [B: set_a,F: nat > a,A: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_644_subset__imageE,axiom,
! [B: set_a,F: complex > a,A: set_complex] :
( ( ord_less_eq_set_a @ B @ ( image_complex_a @ F @ A ) )
=> ~ ! [C3: set_complex] :
( ( ord_le211207098394363844omplex @ C3 @ A )
=> ( B
!= ( image_complex_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_645_subset__imageE,axiom,
! [B: set_a,F: $o > a,A: set_o] :
( ( ord_less_eq_set_a @ B @ ( image_o_a @ F @ A ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
=> ( B
!= ( image_o_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_646_subset__imageE,axiom,
! [B: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_647_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_648_subset__imageE,axiom,
! [B: set_nat,F: complex > nat,A: set_complex] :
( ( ord_less_eq_set_nat @ B @ ( image_complex_nat @ F @ A ) )
=> ~ ! [C3: set_complex] :
( ( ord_le211207098394363844omplex @ C3 @ A )
=> ( B
!= ( image_complex_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_649_subset__imageE,axiom,
! [B: set_nat,F: $o > nat,A: set_o] :
( ( ord_less_eq_set_nat @ B @ ( image_o_nat @ F @ A ) )
=> ~ ! [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
=> ( B
!= ( image_o_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_650_subset__imageE,axiom,
! [B: set_complex,F: a > complex,A: set_a] :
( ( ord_le211207098394363844omplex @ B @ ( image_a_complex @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_a_complex @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_651_subset__imageE,axiom,
! [B: set_complex,F: nat > complex,A: set_nat] :
( ( ord_le211207098394363844omplex @ B @ ( image_nat_complex @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_complex @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_652_image__subsetI,axiom,
! [A: set_nat,F: nat > a,B: set_a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_653_image__subsetI,axiom,
! [A: set_o,F: $o > a,B: set_a] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_o_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_654_image__subsetI,axiom,
! [A: set_a,F: a > a,B: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_655_image__subsetI,axiom,
! [A: set_complex,F: complex > a,B: set_a] :
( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_complex_a @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_656_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_657_image__subsetI,axiom,
! [A: set_o,F: $o > nat,B: set_nat] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_658_image__subsetI,axiom,
! [A: set_a,F: a > nat,B: set_nat] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_659_image__subsetI,axiom,
! [A: set_complex,F: complex > nat,B: set_nat] :
( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_complex_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_660_image__subsetI,axiom,
! [A: set_nat,F: nat > complex,B: set_complex] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_complex @ ( F @ X ) @ B ) )
=> ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_661_image__subsetI,axiom,
! [A: set_o,F: $o > complex,B: set_complex] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( member_complex @ ( F @ X ) @ B ) )
=> ( ord_le211207098394363844omplex @ ( image_o_complex @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_662_image__mono,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_663_image__mono,axiom,
! [A: set_a,B: set_a,F: a > nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_664_image__mono,axiom,
! [A: set_a,B: set_a,F: a > complex] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le211207098394363844omplex @ ( image_a_complex @ F @ A ) @ ( image_a_complex @ F @ B ) ) ) ).
% image_mono
thf(fact_665_image__mono,axiom,
! [A: set_a,B: set_a,F: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_eq_set_o @ ( image_a_o @ F @ A ) @ ( image_a_o @ F @ B ) ) ) ).
% image_mono
thf(fact_666_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ ( image_nat_a @ F @ B ) ) ) ).
% image_mono
thf(fact_667_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_668_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > complex] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ A ) @ ( image_nat_complex @ F @ B ) ) ) ).
% image_mono
thf(fact_669_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A ) @ ( image_nat_o @ F @ B ) ) ) ).
% image_mono
thf(fact_670_image__mono,axiom,
! [A: set_complex,B: set_complex,F: complex > a] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ord_less_eq_set_a @ ( image_complex_a @ F @ A ) @ ( image_complex_a @ F @ B ) ) ) ).
% image_mono
thf(fact_671_image__mono,axiom,
! [A: set_complex,B: set_complex,F: complex > nat] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_complex_nat @ F @ A ) @ ( image_complex_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_672_all__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( P @ ( image_a_a @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_673_all__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_nat_a @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_674_all__subset__image,axiom,
! [F: complex > a,A: set_complex,P: set_a > $o] :
( ( ! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_complex_a @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_complex] :
( ( ord_le211207098394363844omplex @ B2 @ A )
=> ( P @ ( image_complex_a @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_675_all__subset__image,axiom,
! [F: $o > a,A: set_o,P: set_a > $o] :
( ( ! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_o_a @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( P @ ( image_o_a @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_676_all__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( P @ ( image_a_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_677_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_678_all__subset__image,axiom,
! [F: complex > nat,A: set_complex,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_complex_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_complex] :
( ( ord_le211207098394363844omplex @ B2 @ A )
=> ( P @ ( image_complex_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_679_all__subset__image,axiom,
! [F: $o > nat,A: set_o,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_o_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_o] :
( ( ord_less_eq_set_o @ B2 @ A )
=> ( P @ ( image_o_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_680_all__subset__image,axiom,
! [F: a > complex,A: set_a,P: set_complex > $o] :
( ( ! [B2: set_complex] :
( ( ord_le211207098394363844omplex @ B2 @ ( image_a_complex @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( P @ ( image_a_complex @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_681_all__subset__image,axiom,
! [F: nat > complex,A: set_nat,P: set_complex > $o] :
( ( ! [B2: set_complex] :
( ( ord_le211207098394363844omplex @ B2 @ ( image_nat_complex @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_nat_complex @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_682_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > a,B: set_a] :
( ! [X: nat] :
( ( P @ X )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_683_image__Collect__subsetI,axiom,
! [P: complex > $o,F: complex > a,B: set_a] :
( ! [X: complex] :
( ( P @ X )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_complex_a @ F @ ( collect_complex @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_684_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > a,B: set_a] :
( ! [X: a] :
( ( P @ X )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_685_image__Collect__subsetI,axiom,
! [P: $o > $o,F: $o > a,B: set_a] :
( ! [X: $o] :
( ( P @ X )
=> ( member_a @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_o_a @ F @ ( collect_o @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_686_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_687_image__Collect__subsetI,axiom,
! [P: complex > $o,F: complex > nat,B: set_nat] :
( ! [X: complex] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_complex_nat @ F @ ( collect_complex @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_688_image__Collect__subsetI,axiom,
! [P: a > $o,F: a > nat,B: set_nat] :
( ! [X: a] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ ( collect_a @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_689_image__Collect__subsetI,axiom,
! [P: $o > $o,F: $o > nat,B: set_nat] :
( ! [X: $o] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ ( collect_o @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_690_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > complex,B: set_complex] :
( ! [X: nat] :
( ( P @ X )
=> ( member_complex @ ( F @ X ) @ B ) )
=> ( ord_le211207098394363844omplex @ ( image_nat_complex @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_691_image__Collect__subsetI,axiom,
! [P: complex > $o,F: complex > complex,B: set_complex] :
( ! [X: complex] :
( ( P @ X )
=> ( member_complex @ ( F @ X ) @ B ) )
=> ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ F @ ( collect_complex @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_692_pigeonhole__infinite,axiom,
! [A: set_o,F: $o > a] :
( ~ ( finite_finite_o @ A )
=> ( ( finite_finite_a @ ( image_o_a @ F @ A ) )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_693_pigeonhole__infinite,axiom,
! [A: set_o,F: $o > nat] :
( ~ ( finite_finite_o @ A )
=> ( ( finite_finite_nat @ ( image_o_nat @ F @ A ) )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_694_pigeonhole__infinite,axiom,
! [A: set_o,F: $o > complex] :
( ~ ( finite_finite_o @ A )
=> ( ( finite3207457112153483333omplex @ ( image_o_complex @ F @ A ) )
=> ? [X: $o] :
( ( member_o @ X @ A )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_695_pigeonhole__infinite,axiom,
! [A: set_a,F: a > a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_a @ ( image_a_a @ F @ A ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_696_pigeonhole__infinite,axiom,
! [A: set_a,F: a > nat] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_697_pigeonhole__infinite,axiom,
! [A: set_a,F: a > complex] :
( ~ ( finite_finite_a @ A )
=> ( ( finite3207457112153483333omplex @ ( image_a_complex @ F @ A ) )
=> ? [X: a] :
( ( member_a @ X @ A )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_698_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ ( image_nat_a @ F @ A ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_699_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_700_pigeonhole__infinite,axiom,
! [A: set_nat,F: nat > complex] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite3207457112153483333omplex @ ( image_nat_complex @ F @ A ) )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_701_pigeonhole__infinite,axiom,
! [A: set_complex,F: complex > a] :
( ~ ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_a @ ( image_complex_a @ F @ A ) )
=> ? [X: complex] :
( ( member_complex @ X @ A )
& ~ ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [A4: complex] :
( ( member_complex @ A4 @ A )
& ( ( F @ A4 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_702_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: set_nat,B: set_nat] :
( ( ( linord2614967742042102400et_nat @ A )
= ( linord2614967742042102400et_nat @ B ) )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( A = B ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_703_atMost__def,axiom,
( set_ord_atMost_o
= ( ^ [U2: $o] :
( collect_o
@ ^ [X3: $o] : ( ord_less_eq_o @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_704_atMost__def,axiom,
( set_ord_atMost_set_a
= ( ^ [U2: set_a] :
( collect_set_a
@ ^ [X3: set_a] : ( ord_less_eq_set_a @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_705_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_706_atMost__def,axiom,
( set_or9043709113427266269omplex
= ( ^ [U2: set_complex] :
( collect_set_complex
@ ^ [X3: set_complex] : ( ord_le211207098394363844omplex @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_707_atMost__def,axiom,
( set_or6279072120763780779list_a
= ( ^ [U2: set_list_a] :
( collect_set_list_a
@ ^ [X3: set_list_a] : ( ord_le8861187494160871172list_a @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_708_atMost__def,axiom,
( set_ord_atMost_set_o
= ( ^ [U2: set_o] :
( collect_set_o
@ ^ [X3: set_o] : ( ord_less_eq_set_o @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_709_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X3: nat] : ( ord_less_eq_nat @ X3 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_710_finite__distinct__list,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ? [Xs3: list_o] :
( ( ( set_o2 @ Xs3 )
= A )
& ( distinct_o @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_711_finite__distinct__list,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ? [Xs3: list_a] :
( ( ( set_a2 @ Xs3 )
= A )
& ( distinct_a @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_712_finite__distinct__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( ( set_nat2 @ Xs3 )
= A )
& ( distinct_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_713_finite__distinct__list,axiom,
! [A: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ? [Xs3: list_complex] :
( ( ( set_complex2 @ Xs3 )
= A )
& ( distinct_complex @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_714_finite__distinct__list,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ? [Xs3: list_list_a] :
( ( ( set_list_a2 @ Xs3 )
= A )
& ( distinct_list_a @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_715_all__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ! [B2: set_a] :
( ( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_a] :
( ( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ A ) )
=> ( P @ ( image_a_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_716_all__finite__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ! [B2: set_a] :
( ( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_nat_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_717_all__finite__subset__image,axiom,
! [F: complex > a,A: set_complex,P: set_a > $o] :
( ( ! [B2: set_a] :
( ( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_complex_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_complex] :
( ( ( finite3207457112153483333omplex @ B2 )
& ( ord_le211207098394363844omplex @ B2 @ A ) )
=> ( P @ ( image_complex_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_718_all__finite__subset__image,axiom,
! [F: $o > a,A: set_o,P: set_a > $o] :
( ( ! [B2: set_a] :
( ( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_o_a @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_o] :
( ( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ A ) )
=> ( P @ ( image_o_a @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_719_all__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_a] :
( ( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ A ) )
=> ( P @ ( image_a_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_720_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_721_all__finite__subset__image,axiom,
! [F: complex > nat,A: set_complex,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_complex_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_complex] :
( ( ( finite3207457112153483333omplex @ B2 )
& ( ord_le211207098394363844omplex @ B2 @ A ) )
=> ( P @ ( image_complex_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_722_all__finite__subset__image,axiom,
! [F: $o > nat,A: set_o,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_o_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_o] :
( ( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ A ) )
=> ( P @ ( image_o_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_723_all__finite__subset__image,axiom,
! [F: a > complex,A: set_a,P: set_complex > $o] :
( ( ! [B2: set_complex] :
( ( ( finite3207457112153483333omplex @ B2 )
& ( ord_le211207098394363844omplex @ B2 @ ( image_a_complex @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_a] :
( ( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ A ) )
=> ( P @ ( image_a_complex @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_724_all__finite__subset__image,axiom,
! [F: nat > complex,A: set_nat,P: set_complex > $o] :
( ( ! [B2: set_complex] :
( ( ( finite3207457112153483333omplex @ B2 )
& ( ord_le211207098394363844omplex @ B2 @ ( image_nat_complex @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_nat_complex @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_725_ex__finite__subset__image,axiom,
! [F: a > a,A: set_a,P: set_a > $o] :
( ( ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ A )
& ( P @ ( image_a_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_726_ex__finite__subset__image,axiom,
! [F: nat > a,A: set_nat,P: set_a > $o] :
( ( ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_nat_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_727_ex__finite__subset__image,axiom,
! [F: complex > a,A: set_complex,P: set_a > $o] :
( ( ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_complex_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_complex] :
( ( finite3207457112153483333omplex @ B2 )
& ( ord_le211207098394363844omplex @ B2 @ A )
& ( P @ ( image_complex_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_728_ex__finite__subset__image,axiom,
! [F: $o > a,A: set_o,P: set_a > $o] :
( ( ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ ( image_o_a @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_o] :
( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ A )
& ( P @ ( image_o_a @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_729_ex__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ A )
& ( P @ ( image_a_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_730_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_731_ex__finite__subset__image,axiom,
! [F: complex > nat,A: set_complex,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_complex_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_complex] :
( ( finite3207457112153483333omplex @ B2 )
& ( ord_le211207098394363844omplex @ B2 @ A )
& ( P @ ( image_complex_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_732_ex__finite__subset__image,axiom,
! [F: $o > nat,A: set_o,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_o_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_o] :
( ( finite_finite_o @ B2 )
& ( ord_less_eq_set_o @ B2 @ A )
& ( P @ ( image_o_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_733_ex__finite__subset__image,axiom,
! [F: a > complex,A: set_a,P: set_complex > $o] :
( ( ? [B2: set_complex] :
( ( finite3207457112153483333omplex @ B2 )
& ( ord_le211207098394363844omplex @ B2 @ ( image_a_complex @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_a] :
( ( finite_finite_a @ B2 )
& ( ord_less_eq_set_a @ B2 @ A )
& ( P @ ( image_a_complex @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_734_ex__finite__subset__image,axiom,
! [F: nat > complex,A: set_nat,P: set_complex > $o] :
( ( ? [B2: set_complex] :
( ( finite3207457112153483333omplex @ B2 )
& ( ord_le211207098394363844omplex @ B2 @ ( image_nat_complex @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_nat_complex @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_735_finite__subset__image,axiom,
! [B: set_a,F: a > a,A: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
& ( finite_finite_a @ C3 )
& ( B
= ( image_a_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_736_finite__subset__image,axiom,
! [B: set_a,F: nat > a,A: set_nat] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_737_finite__subset__image,axiom,
! [B: set_a,F: complex > a,A: set_complex] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_complex_a @ F @ A ) )
=> ? [C3: set_complex] :
( ( ord_le211207098394363844omplex @ C3 @ A )
& ( finite3207457112153483333omplex @ C3 )
& ( B
= ( image_complex_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_738_finite__subset__image,axiom,
! [B: set_a,F: $o > a,A: set_o] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ ( image_o_a @ F @ A ) )
=> ? [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
& ( finite_finite_o @ C3 )
& ( B
= ( image_o_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_739_finite__subset__image,axiom,
! [B: set_nat,F: a > nat,A: set_a] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
& ( finite_finite_a @ C3 )
& ( B
= ( image_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_740_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_741_finite__subset__image,axiom,
! [B: set_nat,F: complex > nat,A: set_complex] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_complex_nat @ F @ A ) )
=> ? [C3: set_complex] :
( ( ord_le211207098394363844omplex @ C3 @ A )
& ( finite3207457112153483333omplex @ C3 )
& ( B
= ( image_complex_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_742_finite__subset__image,axiom,
! [B: set_nat,F: $o > nat,A: set_o] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_o_nat @ F @ A ) )
=> ? [C3: set_o] :
( ( ord_less_eq_set_o @ C3 @ A )
& ( finite_finite_o @ C3 )
& ( B
= ( image_o_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_743_finite__subset__image,axiom,
! [B: set_complex,F: a > complex,A: set_a] :
( ( finite3207457112153483333omplex @ B )
=> ( ( ord_le211207098394363844omplex @ B @ ( image_a_complex @ F @ A ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
& ( finite_finite_a @ C3 )
& ( B
= ( image_a_complex @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_744_finite__subset__image,axiom,
! [B: set_complex,F: nat > complex,A: set_nat] :
( ( finite3207457112153483333omplex @ B )
=> ( ( ord_le211207098394363844omplex @ B @ ( image_nat_complex @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_complex @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_745_finite__surj,axiom,
! [A: set_a,B: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_746_finite__surj,axiom,
! [A: set_nat,B: set_a,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_747_finite__surj,axiom,
! [A: set_complex,B: set_a,F: complex > a] :
( ( finite3207457112153483333omplex @ A )
=> ( ( ord_less_eq_set_a @ B @ ( image_complex_a @ F @ A ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_surj
thf(fact_748_finite__surj,axiom,
! [A: set_a,B: set_nat,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_749_finite__surj,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_750_finite__surj,axiom,
! [A: set_complex,B: set_nat,F: complex > nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_complex_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_751_finite__surj,axiom,
! [A: set_a,B: set_complex,F: a > complex] :
( ( finite_finite_a @ A )
=> ( ( ord_le211207098394363844omplex @ B @ ( image_a_complex @ F @ A ) )
=> ( finite3207457112153483333omplex @ B ) ) ) ).
% finite_surj
thf(fact_752_finite__surj,axiom,
! [A: set_nat,B: set_complex,F: nat > complex] :
( ( finite_finite_nat @ A )
=> ( ( ord_le211207098394363844omplex @ B @ ( image_nat_complex @ F @ A ) )
=> ( finite3207457112153483333omplex @ B ) ) ) ).
% finite_surj
thf(fact_753_finite__surj,axiom,
! [A: set_complex,B: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( ord_le211207098394363844omplex @ B @ ( image_1468599708987790691omplex @ F @ A ) )
=> ( finite3207457112153483333omplex @ B ) ) ) ).
% finite_surj
thf(fact_754_finite__surj,axiom,
! [A: set_a,B: set_o,F: a > $o] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_o @ B @ ( image_a_o @ F @ A ) )
=> ( finite_finite_o @ B ) ) ) ).
% finite_surj
thf(fact_755_Fpow__mono,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A ) @ ( finite_Fpow_a @ B ) ) ) ).
% Fpow_mono
thf(fact_756_Fpow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_757_Fpow__mono,axiom,
! [A: set_complex,B: set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ord_le4750530260501030778omplex @ ( finite_Fpow_complex @ A ) @ ( finite_Fpow_complex @ B ) ) ) ).
% Fpow_mono
thf(fact_758_Fpow__mono,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B )
=> ( ord_le8877086941679407844list_a @ ( finite_Fpow_list_a @ A ) @ ( finite_Fpow_list_a @ B ) ) ) ).
% Fpow_mono
thf(fact_759_Fpow__mono,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_le4374716579403074808_set_o @ ( finite_Fpow_o @ A ) @ ( finite_Fpow_o @ B ) ) ) ).
% Fpow_mono
thf(fact_760_length__n__lists__elem,axiom,
! [Ys: list_a,N: nat,Xs2: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( n_lists_a @ N @ Xs2 ) ) )
=> ( ( size_size_list_a @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_761_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs2: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_762_length__n__lists__elem,axiom,
! [Ys: list_complex,N: nat,Xs2: list_complex] :
( ( member_list_complex @ Ys @ ( set_list_complex2 @ ( n_lists_complex @ N @ Xs2 ) ) )
=> ( ( size_s3451745648224563538omplex @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_763_length__n__lists__elem,axiom,
! [Ys: list_list_a,N: nat,Xs2: list_list_a] :
( ( member_list_list_a @ Ys @ ( set_list_list_a2 @ ( n_lists_list_a @ N @ Xs2 ) ) )
=> ( ( size_s349497388124573686list_a @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_764_distinct__card,axiom,
! [Xs2: list_o] :
( ( distinct_o @ Xs2 )
=> ( ( finite_card_o @ ( set_o2 @ Xs2 ) )
= ( size_size_list_o @ Xs2 ) ) ) ).
% distinct_card
thf(fact_765_distinct__card,axiom,
! [Xs2: list_list_nat] :
( ( distinct_list_nat @ Xs2 )
=> ( ( finite_card_list_nat @ ( set_list_nat2 @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ).
% distinct_card
thf(fact_766_distinct__card,axiom,
! [Xs2: list_list_complex] :
( ( distin3828661287404608645omplex @ Xs2 )
=> ( ( finite5120063068150530198omplex @ ( set_list_complex2 @ Xs2 ) )
= ( size_s7907857696548412130omplex @ Xs2 ) ) ) ).
% distinct_card
thf(fact_767_distinct__card,axiom,
! [Xs2: list_list_list_a] :
( ( distinct_list_list_a @ Xs2 )
=> ( ( finite9134805042761151410list_a @ ( set_list_list_a2 @ Xs2 ) )
= ( size_s2403821588304063868list_a @ Xs2 ) ) ) ).
% distinct_card
thf(fact_768_distinct__card,axiom,
! [Xs2: list_a] :
( ( distinct_a @ Xs2 )
=> ( ( finite_card_a @ ( set_a2 @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ) ).
% distinct_card
thf(fact_769_distinct__card,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ) ).
% distinct_card
thf(fact_770_distinct__card,axiom,
! [Xs2: list_complex] :
( ( distinct_complex @ Xs2 )
=> ( ( finite_card_complex @ ( set_complex2 @ Xs2 ) )
= ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).
% distinct_card
thf(fact_771_distinct__card,axiom,
! [Xs2: list_list_a] :
( ( distinct_list_a @ Xs2 )
=> ( ( finite_card_list_a @ ( set_list_a2 @ Xs2 ) )
= ( size_s349497388124573686list_a @ Xs2 ) ) ) ).
% distinct_card
thf(fact_772_card__distinct,axiom,
! [Xs2: list_o] :
( ( ( finite_card_o @ ( set_o2 @ Xs2 ) )
= ( size_size_list_o @ Xs2 ) )
=> ( distinct_o @ Xs2 ) ) ).
% card_distinct
thf(fact_773_card__distinct,axiom,
! [Xs2: list_list_nat] :
( ( ( finite_card_list_nat @ ( set_list_nat2 @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( distinct_list_nat @ Xs2 ) ) ).
% card_distinct
thf(fact_774_card__distinct,axiom,
! [Xs2: list_list_complex] :
( ( ( finite5120063068150530198omplex @ ( set_list_complex2 @ Xs2 ) )
= ( size_s7907857696548412130omplex @ Xs2 ) )
=> ( distin3828661287404608645omplex @ Xs2 ) ) ).
% card_distinct
thf(fact_775_card__distinct,axiom,
! [Xs2: list_list_list_a] :
( ( ( finite9134805042761151410list_a @ ( set_list_list_a2 @ Xs2 ) )
= ( size_s2403821588304063868list_a @ Xs2 ) )
=> ( distinct_list_list_a @ Xs2 ) ) ).
% card_distinct
thf(fact_776_card__distinct,axiom,
! [Xs2: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) )
=> ( distinct_a @ Xs2 ) ) ).
% card_distinct
thf(fact_777_card__distinct,axiom,
! [Xs2: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) )
=> ( distinct_nat @ Xs2 ) ) ).
% card_distinct
thf(fact_778_card__distinct,axiom,
! [Xs2: list_complex] :
( ( ( finite_card_complex @ ( set_complex2 @ Xs2 ) )
= ( size_s3451745648224563538omplex @ Xs2 ) )
=> ( distinct_complex @ Xs2 ) ) ).
% card_distinct
thf(fact_779_card__distinct,axiom,
! [Xs2: list_list_a] :
( ( ( finite_card_list_a @ ( set_list_a2 @ Xs2 ) )
= ( size_s349497388124573686list_a @ Xs2 ) )
=> ( distinct_list_a @ Xs2 ) ) ).
% card_distinct
thf(fact_780_card__image__le,axiom,
! [A: set_a,F: a > a] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_a_a @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_781_card__image__le,axiom,
! [A: set_a,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_nat @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_782_card__image__le,axiom,
! [A: set_a,F: a > complex] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_complex @ ( image_a_complex @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_783_card__image__le,axiom,
! [A: set_nat,F: nat > a] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_nat_a @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_784_card__image__le,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_785_card__image__le,axiom,
! [A: set_nat,F: nat > complex] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_complex @ ( image_nat_complex @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_786_card__image__le,axiom,
! [A: set_complex,F: complex > a] :
( ( finite3207457112153483333omplex @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_complex_a @ F @ A ) ) @ ( finite_card_complex @ A ) ) ) ).
% card_image_le
thf(fact_787_card__image__le,axiom,
! [A: set_complex,F: complex > nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_complex_nat @ F @ A ) ) @ ( finite_card_complex @ A ) ) ) ).
% card_image_le
thf(fact_788_card__image__le,axiom,
! [A: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ord_less_eq_nat @ ( finite_card_complex @ ( image_1468599708987790691omplex @ F @ A ) ) @ ( finite_card_complex @ A ) ) ) ).
% card_image_le
thf(fact_789_card__image__le,axiom,
! [A: set_list_nat,F: list_nat > a] :
( ( finite8100373058378681591st_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( image_list_nat_a @ F @ A ) ) @ ( finite_card_list_nat @ A ) ) ) ).
% card_image_le
thf(fact_790_length__n__lists,axiom,
! [N: nat,Xs2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( n_lists_nat @ N @ Xs2 ) )
= ( power_power_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% length_n_lists
thf(fact_791_length__n__lists,axiom,
! [N: nat,Xs2: list_complex] :
( ( size_s7907857696548412130omplex @ ( n_lists_complex @ N @ Xs2 ) )
= ( power_power_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ).
% length_n_lists
thf(fact_792_length__n__lists,axiom,
! [N: nat,Xs2: list_list_a] :
( ( size_s2403821588304063868list_a @ ( n_lists_list_a @ N @ Xs2 ) )
= ( power_power_nat @ ( size_s349497388124573686list_a @ Xs2 ) @ N ) ) ).
% length_n_lists
thf(fact_793_length__n__lists,axiom,
! [N: nat,Xs2: list_a] :
( ( size_s349497388124573686list_a @ ( n_lists_a @ N @ Xs2 ) )
= ( power_power_nat @ ( size_size_list_a @ Xs2 ) @ N ) ) ).
% length_n_lists
thf(fact_794_sum__multicount__gen,axiom,
! [S2: set_o,T4: set_o,R2: $o > $o > $o,K: $o > nat] :
( ( finite_finite_o @ S2 )
=> ( ( finite_finite_o @ T4 )
=> ( ! [X: $o] :
( ( member_o @ X @ T4 )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S2 )
& ( R2 @ I2 @ X ) ) ) )
= ( K @ X ) ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ T4 )
& ( R2 @ I2 @ J ) ) ) )
@ S2 )
= ( groups8507830703676809646_o_nat @ K @ T4 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_795_sum__multicount__gen,axiom,
! [S2: set_o,T4: set_a,R2: $o > a > $o,K: a > nat] :
( ( finite_finite_o @ S2 )
=> ( ( finite_finite_a @ T4 )
=> ( ! [X: a] :
( ( member_a @ X @ T4 )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S2 )
& ( R2 @ I2 @ X ) ) ) )
= ( K @ X ) ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_a
@ ( collect_a
@ ^ [J: a] :
( ( member_a @ J @ T4 )
& ( R2 @ I2 @ J ) ) ) )
@ S2 )
= ( groups6334556678337121940_a_nat @ K @ T4 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_796_sum__multicount__gen,axiom,
! [S2: set_o,T4: set_nat,R2: $o > nat > $o,K: nat > nat] :
( ( finite_finite_o @ S2 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ T4 )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S2 )
& ( R2 @ I2 @ X ) ) ) )
= ( K @ X ) ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ T4 )
& ( R2 @ I2 @ J ) ) ) )
@ S2 )
= ( groups3542108847815614940at_nat @ K @ T4 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_797_sum__multicount__gen,axiom,
! [S2: set_o,T4: set_complex,R2: $o > complex > $o,K: complex > nat] :
( ( finite_finite_o @ S2 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [X: complex] :
( ( member_complex @ X @ T4 )
=> ( ( finite_card_o
@ ( collect_o
@ ^ [I2: $o] :
( ( member_o @ I2 @ S2 )
& ( R2 @ I2 @ X ) ) ) )
= ( K @ X ) ) )
=> ( ( groups8507830703676809646_o_nat
@ ^ [I2: $o] :
( finite_card_complex
@ ( collect_complex
@ ^ [J: complex] :
( ( member_complex @ J @ T4 )
& ( R2 @ I2 @ J ) ) ) )
@ S2 )
= ( groups5693394587270226106ex_nat @ K @ T4 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_798_sum__multicount__gen,axiom,
! [S2: set_a,T4: set_o,R2: a > $o > $o,K: $o > nat] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_o @ T4 )
=> ( ! [X: $o] :
( ( member_o @ X @ T4 )
=> ( ( finite_card_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ S2 )
& ( R2 @ I2 @ X ) ) ) )
= ( K @ X ) ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [I2: a] :
( finite_card_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ T4 )
& ( R2 @ I2 @ J ) ) ) )
@ S2 )
= ( groups8507830703676809646_o_nat @ K @ T4 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_799_sum__multicount__gen,axiom,
! [S2: set_a,T4: set_a,R2: a > a > $o,K: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_a @ T4 )
=> ( ! [X: a] :
( ( member_a @ X @ T4 )
=> ( ( finite_card_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ S2 )
& ( R2 @ I2 @ X ) ) ) )
= ( K @ X ) ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [I2: a] :
( finite_card_a
@ ( collect_a
@ ^ [J: a] :
( ( member_a @ J @ T4 )
& ( R2 @ I2 @ J ) ) ) )
@ S2 )
= ( groups6334556678337121940_a_nat @ K @ T4 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_800_sum__multicount__gen,axiom,
! [S2: set_a,T4: set_nat,R2: a > nat > $o,K: nat > nat] :
( ( finite_finite_a @ S2 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ T4 )
=> ( ( finite_card_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ S2 )
& ( R2 @ I2 @ X ) ) ) )
= ( K @ X ) ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [I2: a] :
( finite_card_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ T4 )
& ( R2 @ I2 @ J ) ) ) )
@ S2 )
= ( groups3542108847815614940at_nat @ K @ T4 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_801_sum__multicount__gen,axiom,
! [S2: set_a,T4: set_complex,R2: a > complex > $o,K: complex > nat] :
( ( finite_finite_a @ S2 )
=> ( ( finite3207457112153483333omplex @ T4 )
=> ( ! [X: complex] :
( ( member_complex @ X @ T4 )
=> ( ( finite_card_a
@ ( collect_a
@ ^ [I2: a] :
( ( member_a @ I2 @ S2 )
& ( R2 @ I2 @ X ) ) ) )
= ( K @ X ) ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [I2: a] :
( finite_card_complex
@ ( collect_complex
@ ^ [J: complex] :
( ( member_complex @ J @ T4 )
& ( R2 @ I2 @ J ) ) ) )
@ S2 )
= ( groups5693394587270226106ex_nat @ K @ T4 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_802_sum__multicount__gen,axiom,
! [S2: set_nat,T4: set_o,R2: nat > $o > $o,K: $o > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_o @ T4 )
=> ( ! [X: $o] :
( ( member_o @ X @ T4 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ S2 )
& ( R2 @ I2 @ X ) ) ) )
= ( K @ X ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( finite_card_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ T4 )
& ( R2 @ I2 @ J ) ) ) )
@ S2 )
= ( groups8507830703676809646_o_nat @ K @ T4 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_803_sum__multicount__gen,axiom,
! [S2: set_nat,T4: set_a,R2: nat > a > $o,K: a > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_a @ T4 )
=> ( ! [X: a] :
( ( member_a @ X @ T4 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ S2 )
& ( R2 @ I2 @ X ) ) ) )
= ( K @ X ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( finite_card_a
@ ( collect_a
@ ^ [J: a] :
( ( member_a @ J @ T4 )
& ( R2 @ I2 @ J ) ) ) )
@ S2 )
= ( groups6334556678337121940_a_nat @ K @ T4 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_804_sum__mono__inv,axiom,
! [F: $o > nat,I3: set_o,G2: $o > nat,I: $o] :
( ( ( groups8507830703676809646_o_nat @ F @ I3 )
= ( groups8507830703676809646_o_nat @ G2 @ I3 ) )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G2 @ I4 ) ) )
=> ( ( member_o @ I @ I3 )
=> ( ( finite_finite_o @ I3 )
=> ( ( F @ I )
= ( G2 @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_805_sum__mono__inv,axiom,
! [F: a > nat,I3: set_a,G2: a > nat,I: a] :
( ( ( groups6334556678337121940_a_nat @ F @ I3 )
= ( groups6334556678337121940_a_nat @ G2 @ I3 ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G2 @ I4 ) ) )
=> ( ( member_a @ I @ I3 )
=> ( ( finite_finite_a @ I3 )
=> ( ( F @ I )
= ( G2 @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_806_sum__mono__inv,axiom,
! [F: nat > nat,I3: set_nat,G2: nat > nat,I: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I3 )
= ( groups3542108847815614940at_nat @ G2 @ I3 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G2 @ I4 ) ) )
=> ( ( member_nat @ I @ I3 )
=> ( ( finite_finite_nat @ I3 )
=> ( ( F @ I )
= ( G2 @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_807_sum__mono__inv,axiom,
! [F: complex > nat,I3: set_complex,G2: complex > nat,I: complex] :
( ( ( groups5693394587270226106ex_nat @ F @ I3 )
= ( groups5693394587270226106ex_nat @ G2 @ I3 ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G2 @ I4 ) ) )
=> ( ( member_complex @ I @ I3 )
=> ( ( finite3207457112153483333omplex @ I3 )
=> ( ( F @ I )
= ( G2 @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_808_sum__mono__inv,axiom,
! [F: list_a > nat,I3: set_list_a,G2: list_a > nat,I: list_a] :
( ( ( groups5521247699297860762_a_nat @ F @ I3 )
= ( groups5521247699297860762_a_nat @ G2 @ I3 ) )
=> ( ! [I4: list_a] :
( ( member_list_a @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G2 @ I4 ) ) )
=> ( ( member_list_a @ I @ I3 )
=> ( ( finite_finite_list_a @ I3 )
=> ( ( F @ I )
= ( G2 @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_809_sum__mono,axiom,
! [K2: set_nat,F: nat > nat,G2: nat > nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G2 @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K2 ) @ ( groups3542108847815614940at_nat @ G2 @ K2 ) ) ) ).
% sum_mono
thf(fact_810_sum__mono,axiom,
! [K2: set_o,F: $o > nat,G2: $o > nat] :
( ! [I4: $o] :
( ( member_o @ I4 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G2 @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K2 ) @ ( groups8507830703676809646_o_nat @ G2 @ K2 ) ) ) ).
% sum_mono
thf(fact_811_sum__mono,axiom,
! [K2: set_a,F: a > nat,G2: a > nat] :
( ! [I4: a] :
( ( member_a @ I4 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G2 @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ K2 ) @ ( groups6334556678337121940_a_nat @ G2 @ K2 ) ) ) ).
% sum_mono
thf(fact_812_sum__mono,axiom,
! [K2: set_complex,F: complex > nat,G2: complex > nat] :
( ! [I4: complex] :
( ( member_complex @ I4 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G2 @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K2 ) @ ( groups5693394587270226106ex_nat @ G2 @ K2 ) ) ) ).
% sum_mono
thf(fact_813_sum__mono,axiom,
! [K2: set_list_a,F: list_a > nat,G2: list_a > nat] :
( ! [I4: list_a] :
( ( member_list_a @ I4 @ K2 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G2 @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups5521247699297860762_a_nat @ F @ K2 ) @ ( groups5521247699297860762_a_nat @ G2 @ K2 ) ) ) ).
% sum_mono
thf(fact_814_finite__nat__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [S3: set_nat] :
? [K3: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_815_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_816_distinct__product__lists,axiom,
! [Xss: list_list_a] :
( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xss ) )
=> ( distinct_a @ X ) )
=> ( distinct_list_a @ ( product_lists_a @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_817_sum__count__set,axiom,
! [Xs2: list_a,X4: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ X4 )
=> ( ( finite_finite_a @ X4 )
=> ( ( groups6334556678337121940_a_nat @ ( count_list_a @ Xs2 ) @ X4 )
= ( size_size_list_a @ Xs2 ) ) ) ) ).
% sum_count_set
thf(fact_818_sum__count__set,axiom,
! [Xs2: list_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ X4 )
=> ( ( finite_finite_nat @ X4 )
=> ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs2 ) @ X4 )
= ( size_size_list_nat @ Xs2 ) ) ) ) ).
% sum_count_set
thf(fact_819_sum__count__set,axiom,
! [Xs2: list_complex,X4: set_complex] :
( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ X4 )
=> ( ( finite3207457112153483333omplex @ X4 )
=> ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs2 ) @ X4 )
= ( size_s3451745648224563538omplex @ Xs2 ) ) ) ) ).
% sum_count_set
thf(fact_820_sum__count__set,axiom,
! [Xs2: list_list_a,X4: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ X4 )
=> ( ( finite_finite_list_a @ X4 )
=> ( ( groups5521247699297860762_a_nat @ ( count_list_list_a @ Xs2 ) @ X4 )
= ( size_s349497388124573686list_a @ Xs2 ) ) ) ) ).
% sum_count_set
thf(fact_821_sum__count__set,axiom,
! [Xs2: list_o,X4: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ X4 )
=> ( ( finite_finite_o @ X4 )
=> ( ( groups8507830703676809646_o_nat @ ( count_list_o @ Xs2 ) @ X4 )
= ( size_size_list_o @ Xs2 ) ) ) ) ).
% sum_count_set
thf(fact_822_in__set__product__lists__length,axiom,
! [Xs2: list_a,Xss: list_list_a] :
( ( member_list_a @ Xs2 @ ( set_list_a2 @ ( product_lists_a @ Xss ) ) )
=> ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_823_in__set__product__lists__length,axiom,
! [Xs2: list_nat,Xss: list_list_nat] :
( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_824_in__set__product__lists__length,axiom,
! [Xs2: list_complex,Xss: list_list_complex] :
( ( member_list_complex @ Xs2 @ ( set_list_complex2 @ ( produc7545014605101902079omplex @ Xss ) ) )
=> ( ( size_s3451745648224563538omplex @ Xs2 )
= ( size_s7907857696548412130omplex @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_825_in__set__product__lists__length,axiom,
! [Xs2: list_list_a,Xss: list_list_list_a] :
( ( member_list_list_a @ Xs2 @ ( set_list_list_a2 @ ( product_lists_list_a @ Xss ) ) )
=> ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s2403821588304063868list_a @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_826_card__UN__le,axiom,
! [I3: set_a,A: a > set_a] :
( ( finite_finite_a @ I3 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ A @ I3 ) ) )
@ ( groups6334556678337121940_a_nat
@ ^ [I2: a] : ( finite_card_a @ ( A @ I2 ) )
@ I3 ) ) ) ).
% card_UN_le
thf(fact_827_card__UN__le,axiom,
! [I3: set_a,A: a > set_complex] :
( ( finite_finite_a @ I3 )
=> ( ord_less_eq_nat @ ( finite_card_complex @ ( comple8424636186594484919omplex @ ( image_a_set_complex @ A @ I3 ) ) )
@ ( groups6334556678337121940_a_nat
@ ^ [I2: a] : ( finite_card_complex @ ( A @ I2 ) )
@ I3 ) ) ) ).
% card_UN_le
thf(fact_828_card__UN__le,axiom,
! [I3: set_nat,A: nat > set_a] :
( ( finite_finite_nat @ I3 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ A @ I3 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( finite_card_a @ ( A @ I2 ) )
@ I3 ) ) ) ).
% card_UN_le
thf(fact_829_card__UN__le,axiom,
! [I3: set_nat,A: nat > set_complex] :
( ( finite_finite_nat @ I3 )
=> ( ord_less_eq_nat @ ( finite_card_complex @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ A @ I3 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( finite_card_complex @ ( A @ I2 ) )
@ I3 ) ) ) ).
% card_UN_le
thf(fact_830_card__UN__le,axiom,
! [I3: set_complex,A: complex > set_a] :
( ( finite3207457112153483333omplex @ I3 )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( comple2307003609928055243_set_a @ ( image_complex_set_a @ A @ I3 ) ) )
@ ( groups5693394587270226106ex_nat
@ ^ [I2: complex] : ( finite_card_a @ ( A @ I2 ) )
@ I3 ) ) ) ).
% card_UN_le
thf(fact_831_card__UN__le,axiom,
! [I3: set_complex,A: complex > set_complex] :
( ( finite3207457112153483333omplex @ I3 )
=> ( ord_less_eq_nat @ ( finite_card_complex @ ( comple8424636186594484919omplex @ ( image_5702600179605932057omplex @ A @ I3 ) ) )
@ ( groups5693394587270226106ex_nat
@ ^ [I2: complex] : ( finite_card_complex @ ( A @ I2 ) )
@ I3 ) ) ) ).
% card_UN_le
thf(fact_832_card__UN__le,axiom,
! [I3: set_a,A: a > set_nat] :
( ( finite_finite_a @ I3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ A @ I3 ) ) )
@ ( groups6334556678337121940_a_nat
@ ^ [I2: a] : ( finite_card_nat @ ( A @ I2 ) )
@ I3 ) ) ) ).
% card_UN_le
thf(fact_833_card__UN__le,axiom,
! [I3: set_nat,A: nat > set_nat] :
( ( finite_finite_nat @ I3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I3 ) ) )
@ ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( finite_card_nat @ ( A @ I2 ) )
@ I3 ) ) ) ).
% card_UN_le
thf(fact_834_card__UN__le,axiom,
! [I3: set_complex,A: complex > set_nat] :
( ( finite3207457112153483333omplex @ I3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( comple7399068483239264473et_nat @ ( image_6352962638927555131et_nat @ A @ I3 ) ) )
@ ( groups5693394587270226106ex_nat
@ ^ [I2: complex] : ( finite_card_nat @ ( A @ I2 ) )
@ I3 ) ) ) ).
% card_UN_le
thf(fact_835_card__UN__le,axiom,
! [I3: set_a,A: a > set_list_a] :
( ( finite_finite_a @ I3 )
=> ( ord_less_eq_nat @ ( finite_card_list_a @ ( comple6928918032620976721list_a @ ( image_a_set_list_a @ A @ I3 ) ) )
@ ( groups6334556678337121940_a_nat
@ ^ [I2: a] : ( finite_card_list_a @ ( A @ I2 ) )
@ I3 ) ) ) ).
% card_UN_le
thf(fact_836_subset__subseqs,axiom,
! [X4: set_a,Xs2: list_a] :
( ( ord_less_eq_set_a @ X4 @ ( set_a2 @ Xs2 ) )
=> ( member_set_a @ X4 @ ( image_list_a_set_a @ set_a2 @ ( set_list_a2 @ ( subseqs_a @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_837_subset__subseqs,axiom,
! [X4: set_nat,Xs2: list_nat] :
( ( ord_less_eq_set_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( member_set_nat @ X4 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_838_subset__subseqs,axiom,
! [X4: set_complex,Xs2: list_complex] :
( ( ord_le211207098394363844omplex @ X4 @ ( set_complex2 @ Xs2 ) )
=> ( member_set_complex @ X4 @ ( image_1532835712220217129omplex @ set_complex2 @ ( set_list_complex2 @ ( subseqs_complex @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_839_subset__subseqs,axiom,
! [X4: set_list_a,Xs2: list_list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
=> ( member_set_list_a @ X4 @ ( image_432481560377026271list_a @ set_list_a2 @ ( set_list_list_a2 @ ( subseqs_list_a @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_840_subset__subseqs,axiom,
! [X4: set_o,Xs2: list_o] :
( ( ord_less_eq_set_o @ X4 @ ( set_o2 @ Xs2 ) )
=> ( member_set_o @ X4 @ ( image_list_o_set_o @ set_o2 @ ( set_list_o2 @ ( subseqs_o @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_841_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A ) )
= ( Sup @ A ) ) ).
% Sup.SUP_identity_eq
thf(fact_842_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A ) )
= ( Inf @ A ) ) ).
% Inf.INF_identity_eq
thf(fact_843_inj__on__iff__card__le,axiom,
! [A: set_a,B: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ? [F3: a > a] :
( ( inj_on_a_a @ F3 @ A )
& ( ord_less_eq_set_a @ ( image_a_a @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_844_inj__on__iff__card__le,axiom,
! [A: set_nat,B: set_a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ? [F3: nat > a] :
( ( inj_on_nat_a @ F3 @ A )
& ( ord_less_eq_set_a @ ( image_nat_a @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_845_inj__on__iff__card__le,axiom,
! [A: set_complex,B: set_a] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_a @ B )
=> ( ( ? [F3: complex > a] :
( ( inj_on_complex_a @ F3 @ A )
& ( ord_less_eq_set_a @ ( image_complex_a @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_a @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_846_inj__on__iff__card__le,axiom,
! [A: set_a,B: set_nat] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ? [F3: a > nat] :
( ( inj_on_a_nat @ F3 @ A )
& ( ord_less_eq_set_nat @ ( image_a_nat @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_847_inj__on__iff__card__le,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ? [F3: nat > nat] :
( ( inj_on_nat_nat @ F3 @ A )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_848_inj__on__iff__card__le,axiom,
! [A: set_complex,B: set_nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ? [F3: complex > nat] :
( ( inj_on_complex_nat @ F3 @ A )
& ( ord_less_eq_set_nat @ ( image_complex_nat @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_849_inj__on__iff__card__le,axiom,
! [A: set_a,B: set_complex] :
( ( finite_finite_a @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ( ? [F3: a > complex] :
( ( inj_on_a_complex @ F3 @ A )
& ( ord_le211207098394363844omplex @ ( image_a_complex @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_complex @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_850_inj__on__iff__card__le,axiom,
! [A: set_nat,B: set_complex] :
( ( finite_finite_nat @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ( ? [F3: nat > complex] :
( ( inj_on_nat_complex @ F3 @ A )
& ( ord_le211207098394363844omplex @ ( image_nat_complex @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_complex @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_851_inj__on__iff__card__le,axiom,
! [A: set_complex,B: set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite3207457112153483333omplex @ B )
=> ( ( ? [F3: complex > complex] :
( ( inj_on2498852929715845839omplex @ F3 @ A )
& ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_complex @ A ) @ ( finite_card_complex @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_852_inj__on__iff__card__le,axiom,
! [A: set_a,B: set_o] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_o @ B )
=> ( ( ? [F3: a > $o] :
( ( inj_on_a_o @ F3 @ A )
& ( ord_less_eq_set_o @ ( image_a_o @ F3 @ A ) @ B ) ) )
= ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_o @ B ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_853_UN__ball__bex__simps_I3_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ A ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ? [Y4: nat] :
( ( member_nat @ Y4 @ X3 )
& ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_854_UN__ball__bex__simps_I1_J,axiom,
! [A: set_set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ A ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ X3 )
=> ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_855_UnionI,axiom,
! [X4: set_o,C2: set_set_o,A: $o] :
( ( member_set_o @ X4 @ C2 )
=> ( ( member_o @ A @ X4 )
=> ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) ) ) ) ).
% UnionI
thf(fact_856_UnionI,axiom,
! [X4: set_a,C2: set_set_a,A: a] :
( ( member_set_a @ X4 @ C2 )
=> ( ( member_a @ A @ X4 )
=> ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) ) ) ) ).
% UnionI
thf(fact_857_UnionI,axiom,
! [X4: set_complex,C2: set_set_complex,A: complex] :
( ( member_set_complex @ X4 @ C2 )
=> ( ( member_complex @ A @ X4 )
=> ( member_complex @ A @ ( comple8424636186594484919omplex @ C2 ) ) ) ) ).
% UnionI
thf(fact_858_UnionI,axiom,
! [X4: set_list_a,C2: set_set_list_a,A: list_a] :
( ( member_set_list_a @ X4 @ C2 )
=> ( ( member_list_a @ A @ X4 )
=> ( member_list_a @ A @ ( comple6928918032620976721list_a @ C2 ) ) ) ) ).
% UnionI
thf(fact_859_UnionI,axiom,
! [X4: set_nat,C2: set_set_nat,A: nat] :
( ( member_set_nat @ X4 @ C2 )
=> ( ( member_nat @ A @ X4 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_860_Union__iff,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
= ( ? [X3: set_o] :
( ( member_set_o @ X3 @ C2 )
& ( member_o @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_861_Union__iff,axiom,
! [A: a,C2: set_set_a] :
( ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) )
= ( ? [X3: set_a] :
( ( member_set_a @ X3 @ C2 )
& ( member_a @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_862_Union__iff,axiom,
! [A: complex,C2: set_set_complex] :
( ( member_complex @ A @ ( comple8424636186594484919omplex @ C2 ) )
= ( ? [X3: set_complex] :
( ( member_set_complex @ X3 @ C2 )
& ( member_complex @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_863_Union__iff,axiom,
! [A: list_a,C2: set_set_list_a] :
( ( member_list_a @ A @ ( comple6928918032620976721list_a @ C2 ) )
= ( ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ C2 )
& ( member_list_a @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_864_Union__iff,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ C2 )
& ( member_nat @ A @ X3 ) ) ) ) ).
% Union_iff
thf(fact_865_Sup__atMost,axiom,
! [Y: $o] :
( ( complete_Sup_Sup_o @ ( set_ord_atMost_o @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_866_Sup__atMost,axiom,
! [Y: set_nat] :
( ( comple7399068483239264473et_nat @ ( set_or4236626031148496127et_nat @ Y ) )
= Y ) ).
% Sup_atMost
thf(fact_867_ball__UN,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( B @ X3 ) )
=> ( P @ Y4 ) ) ) ) ) ).
% ball_UN
thf(fact_868_bex__UN,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ? [Y4: nat] :
( ( member_nat @ Y4 @ ( B @ X3 ) )
& ( P @ Y4 ) ) ) ) ) ).
% bex_UN
thf(fact_869_UN__ball__bex__simps_I2_J,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( B @ X3 ) )
=> ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_870_UN__ball__bex__simps_I4_J,axiom,
! [B: nat > set_nat,A: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ? [Y4: nat] :
( ( member_nat @ Y4 @ ( B @ X3 ) )
& ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_871_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X3: $o] : X3
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_872_SUP__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_873_SUP__identity__eq,axiom,
! [A: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [X3: set_nat] : X3
@ A ) )
= ( comple7399068483239264473et_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_874_UN__iff,axiom,
! [B3: nat,B: nat > set_nat,A: set_nat] :
( ( member_nat @ B3 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( member_nat @ B3 @ ( B @ X3 ) ) ) ) ) ).
% UN_iff
thf(fact_875_UN__I,axiom,
! [A2: nat,A: set_nat,B3: $o,B: nat > set_o] :
( ( member_nat @ A2 @ A )
=> ( ( member_o @ B3 @ ( B @ A2 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_876_UN__I,axiom,
! [A2: nat,A: set_nat,B3: a,B: nat > set_a] :
( ( member_nat @ A2 @ A )
=> ( ( member_a @ B3 @ ( B @ A2 ) )
=> ( member_a @ B3 @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_877_UN__I,axiom,
! [A2: nat,A: set_nat,B3: complex,B: nat > set_complex] :
( ( member_nat @ A2 @ A )
=> ( ( member_complex @ B3 @ ( B @ A2 ) )
=> ( member_complex @ B3 @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_878_UN__I,axiom,
! [A2: $o,A: set_o,B3: $o,B: $o > set_o] :
( ( member_o @ A2 @ A )
=> ( ( member_o @ B3 @ ( B @ A2 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_879_UN__I,axiom,
! [A2: $o,A: set_o,B3: a,B: $o > set_a] :
( ( member_o @ A2 @ A )
=> ( ( member_a @ B3 @ ( B @ A2 ) )
=> ( member_a @ B3 @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_880_UN__I,axiom,
! [A2: $o,A: set_o,B3: complex,B: $o > set_complex] :
( ( member_o @ A2 @ A )
=> ( ( member_complex @ B3 @ ( B @ A2 ) )
=> ( member_complex @ B3 @ ( comple8424636186594484919omplex @ ( image_o_set_complex @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_881_UN__I,axiom,
! [A2: a,A: set_a,B3: $o,B: a > set_o] :
( ( member_a @ A2 @ A )
=> ( ( member_o @ B3 @ ( B @ A2 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_882_UN__I,axiom,
! [A2: a,A: set_a,B3: a,B: a > set_a] :
( ( member_a @ A2 @ A )
=> ( ( member_a @ B3 @ ( B @ A2 ) )
=> ( member_a @ B3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_883_UN__I,axiom,
! [A2: a,A: set_a,B3: complex,B: a > set_complex] :
( ( member_a @ A2 @ A )
=> ( ( member_complex @ B3 @ ( B @ A2 ) )
=> ( member_complex @ B3 @ ( comple8424636186594484919omplex @ ( image_a_set_complex @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_884_UN__I,axiom,
! [A2: complex,A: set_complex,B3: $o,B: complex > set_o] :
( ( member_complex @ A2 @ A )
=> ( ( member_o @ B3 @ ( B @ A2 ) )
=> ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_complex_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_885_finite__UN,axiom,
! [A: set_a,B: a > set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( finite_finite_a @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_886_finite__UN,axiom,
! [A: set_a,B: a > set_complex] :
( ( finite_finite_a @ A )
=> ( ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ ( image_a_set_complex @ B @ A ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( finite3207457112153483333omplex @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_887_finite__UN,axiom,
! [A: set_nat,B: nat > set_a] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( finite_finite_a @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_888_finite__UN,axiom,
! [A: set_nat,B: nat > set_complex] :
( ( finite_finite_nat @ A )
=> ( ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ B @ A ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( finite3207457112153483333omplex @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_889_finite__UN,axiom,
! [A: set_complex,B: complex > set_a] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_complex_set_a @ B @ A ) ) )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( finite_finite_a @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_890_finite__UN,axiom,
! [A: set_complex,B: complex > set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ ( image_5702600179605932057omplex @ B @ A ) ) )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( finite3207457112153483333omplex @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_891_finite__UN,axiom,
! [A: set_a,B: a > set_nat] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( finite_finite_nat @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_892_finite__UN,axiom,
! [A: set_nat,B: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( finite_finite_nat @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_893_finite__UN,axiom,
! [A: set_complex,B: complex > set_nat] :
( ( finite3207457112153483333omplex @ A )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_6352962638927555131et_nat @ B @ A ) ) )
= ( ! [X3: complex] :
( ( member_complex @ X3 @ A )
=> ( finite_finite_nat @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_894_finite__UN,axiom,
! [A: set_a,B: a > set_list_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_a_set_list_a @ B @ A ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( finite_finite_list_a @ ( B @ X3 ) ) ) ) ) ) ).
% finite_UN
thf(fact_895_finite__Union,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ! [M4: set_a] :
( ( member_set_a @ M4 @ A )
=> ( finite_finite_a @ M4 ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% finite_Union
thf(fact_896_finite__Union,axiom,
! [A: set_set_complex] :
( ( finite6551019134538273531omplex @ A )
=> ( ! [M4: set_complex] :
( ( member_set_complex @ M4 @ A )
=> ( finite3207457112153483333omplex @ M4 ) )
=> ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ A ) ) ) ) ).
% finite_Union
thf(fact_897_finite__Union,axiom,
! [A: set_set_list_a] :
( ( finite5282473924520328461list_a @ A )
=> ( ! [M4: set_list_a] :
( ( member_set_list_a @ M4 @ A )
=> ( finite_finite_list_a @ M4 ) )
=> ( finite_finite_list_a @ ( comple6928918032620976721list_a @ A ) ) ) ) ).
% finite_Union
thf(fact_898_finite__Union,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [M4: set_nat] :
( ( member_set_nat @ M4 @ A )
=> ( finite_finite_nat @ M4 ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% finite_Union
thf(fact_899_finite__UN__I,axiom,
! [A: set_o,B: $o > set_a] :
( ( finite_finite_o @ A )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ( finite_finite_a @ ( B @ A5 ) ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_900_finite__UN__I,axiom,
! [A: set_o,B: $o > set_complex] :
( ( finite_finite_o @ A )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ( finite3207457112153483333omplex @ ( B @ A5 ) ) )
=> ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ ( image_o_set_complex @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_901_finite__UN__I,axiom,
! [A: set_a,B: a > set_a] :
( ( finite_finite_a @ A )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ( finite_finite_a @ ( B @ A5 ) ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_902_finite__UN__I,axiom,
! [A: set_a,B: a > set_complex] :
( ( finite_finite_a @ A )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ( finite3207457112153483333omplex @ ( B @ A5 ) ) )
=> ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ ( image_a_set_complex @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_903_finite__UN__I,axiom,
! [A: set_nat,B: nat > set_a] :
( ( finite_finite_nat @ A )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ( finite_finite_a @ ( B @ A5 ) ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_904_finite__UN__I,axiom,
! [A: set_nat,B: nat > set_complex] :
( ( finite_finite_nat @ A )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ( finite3207457112153483333omplex @ ( B @ A5 ) ) )
=> ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_905_finite__UN__I,axiom,
! [A: set_complex,B: complex > set_a] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [A5: complex] :
( ( member_complex @ A5 @ A )
=> ( finite_finite_a @ ( B @ A5 ) ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_complex_set_a @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_906_finite__UN__I,axiom,
! [A: set_complex,B: complex > set_complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ! [A5: complex] :
( ( member_complex @ A5 @ A )
=> ( finite3207457112153483333omplex @ ( B @ A5 ) ) )
=> ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ ( image_5702600179605932057omplex @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_907_finite__UN__I,axiom,
! [A: set_o,B: $o > set_nat] :
( ( finite_finite_o @ A )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ( finite_finite_nat @ ( B @ A5 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_908_finite__UN__I,axiom,
! [A: set_a,B: a > set_nat] :
( ( finite_finite_a @ A )
=> ( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ( finite_finite_nat @ ( B @ A5 ) ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B @ A ) ) ) ) ) ).
% finite_UN_I
thf(fact_909_inj__on__image,axiom,
! [F: nat > set_nat,A: set_set_nat] :
( ( inj_on_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on2776966659131765557et_nat @ ( image_nat_set_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_910_inj__on__image,axiom,
! [F: nat > nat,A: set_set_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_911_Sup__upper2,axiom,
! [U: set_a,A: set_set_a,V: set_a] :
( ( member_set_a @ U @ A )
=> ( ( ord_less_eq_set_a @ V @ U )
=> ( ord_less_eq_set_a @ V @ ( comple2307003609928055243_set_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_912_Sup__upper2,axiom,
! [U: set_complex,A: set_set_complex,V: set_complex] :
( ( member_set_complex @ U @ A )
=> ( ( ord_le211207098394363844omplex @ V @ U )
=> ( ord_le211207098394363844omplex @ V @ ( comple8424636186594484919omplex @ A ) ) ) ) ).
% Sup_upper2
thf(fact_913_Sup__upper2,axiom,
! [U: set_list_a,A: set_set_list_a,V: set_list_a] :
( ( member_set_list_a @ U @ A )
=> ( ( ord_le8861187494160871172list_a @ V @ U )
=> ( ord_le8861187494160871172list_a @ V @ ( comple6928918032620976721list_a @ A ) ) ) ) ).
% Sup_upper2
thf(fact_914_Sup__upper2,axiom,
! [U: set_o,A: set_set_o,V: set_o] :
( ( member_set_o @ U @ A )
=> ( ( ord_less_eq_set_o @ V @ U )
=> ( ord_less_eq_set_o @ V @ ( comple90263536869209701_set_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_915_Sup__upper2,axiom,
! [U: $o,A: set_o,V: $o] :
( ( member_o @ U @ A )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_916_Sup__upper2,axiom,
! [U: set_nat,A: set_set_nat,V: set_nat] :
( ( member_set_nat @ U @ A )
=> ( ( ord_less_eq_set_nat @ V @ U )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_917_Sup__le__iff,axiom,
! [A: set_set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ B3 )
= ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ord_less_eq_set_a @ X3 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_918_Sup__le__iff,axiom,
! [A: set_set_complex,B3: set_complex] :
( ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ A ) @ B3 )
= ( ! [X3: set_complex] :
( ( member_set_complex @ X3 @ A )
=> ( ord_le211207098394363844omplex @ X3 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_919_Sup__le__iff,axiom,
! [A: set_set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( comple6928918032620976721list_a @ A ) @ B3 )
= ( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A )
=> ( ord_le8861187494160871172list_a @ X3 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_920_Sup__le__iff,axiom,
! [A: set_set_o,B3: set_o] :
( ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ B3 )
= ( ! [X3: set_o] :
( ( member_set_o @ X3 @ A )
=> ( ord_less_eq_set_o @ X3 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_921_Sup__le__iff,axiom,
! [A: set_o,B3: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B3 )
= ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ X3 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_922_Sup__le__iff,axiom,
! [A: set_set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ B3 )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ B3 ) ) ) ) ).
% Sup_le_iff
thf(fact_923_Sup__upper,axiom,
! [X2: set_a,A: set_set_a] :
( ( member_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Sup_upper
thf(fact_924_Sup__upper,axiom,
! [X2: set_complex,A: set_set_complex] :
( ( member_set_complex @ X2 @ A )
=> ( ord_le211207098394363844omplex @ X2 @ ( comple8424636186594484919omplex @ A ) ) ) ).
% Sup_upper
thf(fact_925_Sup__upper,axiom,
! [X2: set_list_a,A: set_set_list_a] :
( ( member_set_list_a @ X2 @ A )
=> ( ord_le8861187494160871172list_a @ X2 @ ( comple6928918032620976721list_a @ A ) ) ) ).
% Sup_upper
thf(fact_926_Sup__upper,axiom,
! [X2: set_o,A: set_set_o] :
( ( member_set_o @ X2 @ A )
=> ( ord_less_eq_set_o @ X2 @ ( comple90263536869209701_set_o @ A ) ) ) ).
% Sup_upper
thf(fact_927_Sup__upper,axiom,
! [X2: $o,A: set_o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_928_Sup__upper,axiom,
! [X2: set_nat,A: set_set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_929_Sup__least,axiom,
! [A: set_set_a,Z3: set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ Z3 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ Z3 ) ) ).
% Sup_least
thf(fact_930_Sup__least,axiom,
! [A: set_set_complex,Z3: set_complex] :
( ! [X: set_complex] :
( ( member_set_complex @ X @ A )
=> ( ord_le211207098394363844omplex @ X @ Z3 ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ A ) @ Z3 ) ) ).
% Sup_least
thf(fact_931_Sup__least,axiom,
! [A: set_set_list_a,Z3: set_list_a] :
( ! [X: set_list_a] :
( ( member_set_list_a @ X @ A )
=> ( ord_le8861187494160871172list_a @ X @ Z3 ) )
=> ( ord_le8861187494160871172list_a @ ( comple6928918032620976721list_a @ A ) @ Z3 ) ) ).
% Sup_least
thf(fact_932_Sup__least,axiom,
! [A: set_set_o,Z3: set_o] :
( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( ord_less_eq_set_o @ X @ Z3 ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ Z3 ) ) ).
% Sup_least
thf(fact_933_Sup__least,axiom,
! [A: set_o,Z3: $o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ Z3 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z3 ) ) ).
% Sup_least
thf(fact_934_Sup__least,axiom,
! [A: set_set_nat,Z3: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ Z3 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ Z3 ) ) ).
% Sup_least
thf(fact_935_Sup__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [A5: set_a] :
( ( member_set_a @ A5 @ A )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ B )
& ( ord_less_eq_set_a @ A5 @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_mono
thf(fact_936_Sup__mono,axiom,
! [A: set_set_complex,B: set_set_complex] :
( ! [A5: set_complex] :
( ( member_set_complex @ A5 @ A )
=> ? [X5: set_complex] :
( ( member_set_complex @ X5 @ B )
& ( ord_le211207098394363844omplex @ A5 @ X5 ) ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ A ) @ ( comple8424636186594484919omplex @ B ) ) ) ).
% Sup_mono
thf(fact_937_Sup__mono,axiom,
! [A: set_set_list_a,B: set_set_list_a] :
( ! [A5: set_list_a] :
( ( member_set_list_a @ A5 @ A )
=> ? [X5: set_list_a] :
( ( member_set_list_a @ X5 @ B )
& ( ord_le8861187494160871172list_a @ A5 @ X5 ) ) )
=> ( ord_le8861187494160871172list_a @ ( comple6928918032620976721list_a @ A ) @ ( comple6928918032620976721list_a @ B ) ) ) ).
% Sup_mono
thf(fact_938_Sup__mono,axiom,
! [A: set_set_o,B: set_set_o] :
( ! [A5: set_o] :
( ( member_set_o @ A5 @ A )
=> ? [X5: set_o] :
( ( member_set_o @ X5 @ B )
& ( ord_less_eq_set_o @ A5 @ X5 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ ( comple90263536869209701_set_o @ B ) ) ) ).
% Sup_mono
thf(fact_939_Sup__mono,axiom,
! [A: set_o,B: set_o] :
( ! [A5: $o] :
( ( member_o @ A5 @ A )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ A5 @ X5 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_940_Sup__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ A5 @ X5 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_mono
thf(fact_941_Sup__eqI,axiom,
! [A: set_set_a,X2: set_a] :
( ! [Y2: set_a] :
( ( member_set_a @ Y2 @ A )
=> ( ord_less_eq_set_a @ Y2 @ X2 ) )
=> ( ! [Y2: set_a] :
( ! [Z4: set_a] :
( ( member_set_a @ Z4 @ A )
=> ( ord_less_eq_set_a @ Z4 @ Y2 ) )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) )
=> ( ( comple2307003609928055243_set_a @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_942_Sup__eqI,axiom,
! [A: set_set_complex,X2: set_complex] :
( ! [Y2: set_complex] :
( ( member_set_complex @ Y2 @ A )
=> ( ord_le211207098394363844omplex @ Y2 @ X2 ) )
=> ( ! [Y2: set_complex] :
( ! [Z4: set_complex] :
( ( member_set_complex @ Z4 @ A )
=> ( ord_le211207098394363844omplex @ Z4 @ Y2 ) )
=> ( ord_le211207098394363844omplex @ X2 @ Y2 ) )
=> ( ( comple8424636186594484919omplex @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_943_Sup__eqI,axiom,
! [A: set_set_list_a,X2: set_list_a] :
( ! [Y2: set_list_a] :
( ( member_set_list_a @ Y2 @ A )
=> ( ord_le8861187494160871172list_a @ Y2 @ X2 ) )
=> ( ! [Y2: set_list_a] :
( ! [Z4: set_list_a] :
( ( member_set_list_a @ Z4 @ A )
=> ( ord_le8861187494160871172list_a @ Z4 @ Y2 ) )
=> ( ord_le8861187494160871172list_a @ X2 @ Y2 ) )
=> ( ( comple6928918032620976721list_a @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_944_Sup__eqI,axiom,
! [A: set_set_o,X2: set_o] :
( ! [Y2: set_o] :
( ( member_set_o @ Y2 @ A )
=> ( ord_less_eq_set_o @ Y2 @ X2 ) )
=> ( ! [Y2: set_o] :
( ! [Z4: set_o] :
( ( member_set_o @ Z4 @ A )
=> ( ord_less_eq_set_o @ Z4 @ Y2 ) )
=> ( ord_less_eq_set_o @ X2 @ Y2 ) )
=> ( ( comple90263536869209701_set_o @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_945_Sup__eqI,axiom,
! [A: set_o,X2: $o] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ A )
=> ( ord_less_eq_o @ Y2 @ X2 ) )
=> ( ! [Y2: $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ A )
=> ( ord_less_eq_o @ Z4 @ Y2 ) )
=> ( ord_less_eq_o @ X2 @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_946_Sup__eqI,axiom,
! [A: set_set_nat,X2: set_nat] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ A )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ! [Y2: set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ A )
=> ( ord_less_eq_set_nat @ Z4 @ Y2 ) )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) )
=> ( ( comple7399068483239264473et_nat @ A )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_947_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > $o,D: nat > $o] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_948_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > $o,D: $o > $o] :
( ( A = B )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_949_SUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > $o,D: a > $o] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_950_SUP__cong,axiom,
! [A: set_complex,B: set_complex,C2: complex > $o,D: complex > $o] :
( ( A = B )
=> ( ! [X: complex] :
( ( member_complex @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_complex_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_complex_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_951_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D: nat > nat] :
( ( A = B )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_952_SUP__cong,axiom,
! [A: set_o,B: set_o,C2: $o > nat,D: $o > nat] :
( ( A = B )
=> ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_953_SUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > nat,D: a > nat] :
( ( A = B )
=> ( ! [X: a] :
( ( member_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_a_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_a_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_954_SUP__cong,axiom,
! [A: set_complex,B: set_complex,C2: complex > nat,D: complex > nat] :
( ( A = B )
=> ( ! [X: complex] :
( ( member_complex @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_complex_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_complex_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_955_SUP__cong,axiom,
! [A: set_list_a,B: set_list_a,C2: list_a > $o,D: list_a > $o] :
( ( A = B )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_list_a_o @ C2 @ A ) )
= ( complete_Sup_Sup_o @ ( image_list_a_o @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_956_SUP__cong,axiom,
! [A: set_list_a,B: set_list_a,C2: list_a > nat,D: list_a > nat] :
( ( A = B )
=> ( ! [X: list_a] :
( ( member_list_a @ X @ B )
=> ( ( C2 @ X )
= ( D @ X ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_list_a_nat @ C2 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_list_a_nat @ D @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_957_Union__least,axiom,
! [A: set_set_a,C2: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A )
=> ( ord_less_eq_set_a @ X7 @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_958_Union__least,axiom,
! [A: set_set_complex,C2: set_complex] :
( ! [X7: set_complex] :
( ( member_set_complex @ X7 @ A )
=> ( ord_le211207098394363844omplex @ X7 @ C2 ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ A ) @ C2 ) ) ).
% Union_least
thf(fact_959_Union__least,axiom,
! [A: set_set_list_a,C2: set_list_a] :
( ! [X7: set_list_a] :
( ( member_set_list_a @ X7 @ A )
=> ( ord_le8861187494160871172list_a @ X7 @ C2 ) )
=> ( ord_le8861187494160871172list_a @ ( comple6928918032620976721list_a @ A ) @ C2 ) ) ).
% Union_least
thf(fact_960_Union__least,axiom,
! [A: set_set_o,C2: set_o] :
( ! [X7: set_o] :
( ( member_set_o @ X7 @ A )
=> ( ord_less_eq_set_o @ X7 @ C2 ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ C2 ) ) ).
% Union_least
thf(fact_961_Union__least,axiom,
! [A: set_set_nat,C2: set_nat] :
( ! [X7: set_nat] :
( ( member_set_nat @ X7 @ A )
=> ( ord_less_eq_set_nat @ X7 @ C2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ C2 ) ) ).
% Union_least
thf(fact_962_Union__upper,axiom,
! [B: set_a,A: set_set_a] :
( ( member_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ ( comple2307003609928055243_set_a @ A ) ) ) ).
% Union_upper
thf(fact_963_Union__upper,axiom,
! [B: set_complex,A: set_set_complex] :
( ( member_set_complex @ B @ A )
=> ( ord_le211207098394363844omplex @ B @ ( comple8424636186594484919omplex @ A ) ) ) ).
% Union_upper
thf(fact_964_Union__upper,axiom,
! [B: set_list_a,A: set_set_list_a] :
( ( member_set_list_a @ B @ A )
=> ( ord_le8861187494160871172list_a @ B @ ( comple6928918032620976721list_a @ A ) ) ) ).
% Union_upper
thf(fact_965_Union__upper,axiom,
! [B: set_o,A: set_set_o] :
( ( member_set_o @ B @ A )
=> ( ord_less_eq_set_o @ B @ ( comple90263536869209701_set_o @ A ) ) ) ).
% Union_upper
thf(fact_966_Union__upper,axiom,
! [B: set_nat,A: set_set_nat] :
( ( member_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Union_upper
thf(fact_967_Union__subsetI,axiom,
! [A: set_set_a,B: set_set_a] :
( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ? [Y5: set_a] :
( ( member_set_a @ Y5 @ B )
& ( ord_less_eq_set_a @ X @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_subsetI
thf(fact_968_Union__subsetI,axiom,
! [A: set_set_complex,B: set_set_complex] :
( ! [X: set_complex] :
( ( member_set_complex @ X @ A )
=> ? [Y5: set_complex] :
( ( member_set_complex @ Y5 @ B )
& ( ord_le211207098394363844omplex @ X @ Y5 ) ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ A ) @ ( comple8424636186594484919omplex @ B ) ) ) ).
% Union_subsetI
thf(fact_969_Union__subsetI,axiom,
! [A: set_set_list_a,B: set_set_list_a] :
( ! [X: set_list_a] :
( ( member_set_list_a @ X @ A )
=> ? [Y5: set_list_a] :
( ( member_set_list_a @ Y5 @ B )
& ( ord_le8861187494160871172list_a @ X @ Y5 ) ) )
=> ( ord_le8861187494160871172list_a @ ( comple6928918032620976721list_a @ A ) @ ( comple6928918032620976721list_a @ B ) ) ) ).
% Union_subsetI
thf(fact_970_Union__subsetI,axiom,
! [A: set_set_o,B: set_set_o] :
( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ? [Y5: set_o] :
( ( member_set_o @ Y5 @ B )
& ( ord_less_eq_set_o @ X @ Y5 ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ ( comple90263536869209701_set_o @ B ) ) ) ).
% Union_subsetI
thf(fact_971_Union__subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ? [Y5: set_nat] :
( ( member_set_nat @ Y5 @ B )
& ( ord_less_eq_set_nat @ X @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_972_UnionE,axiom,
! [A: $o,C2: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C2 ) )
=> ~ ! [X7: set_o] :
( ( member_o @ A @ X7 )
=> ~ ( member_set_o @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_973_UnionE,axiom,
! [A: a,C2: set_set_a] :
( ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) )
=> ~ ! [X7: set_a] :
( ( member_a @ A @ X7 )
=> ~ ( member_set_a @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_974_UnionE,axiom,
! [A: complex,C2: set_set_complex] :
( ( member_complex @ A @ ( comple8424636186594484919omplex @ C2 ) )
=> ~ ! [X7: set_complex] :
( ( member_complex @ A @ X7 )
=> ~ ( member_set_complex @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_975_UnionE,axiom,
! [A: list_a,C2: set_set_list_a] :
( ( member_list_a @ A @ ( comple6928918032620976721list_a @ C2 ) )
=> ~ ! [X7: set_list_a] :
( ( member_list_a @ A @ X7 )
=> ~ ( member_set_list_a @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_976_UnionE,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X7: set_nat] :
( ( member_nat @ A @ X7 )
=> ~ ( member_set_nat @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_977_inj__on__UNION__chain,axiom,
! [I3: set_nat,A: nat > set_nat,F: nat > nat] :
( ! [I4: nat,J2: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ( member_nat @ J2 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A @ I4 ) @ ( A @ J2 ) )
| ( ord_less_eq_set_nat @ ( A @ J2 ) @ ( A @ I4 ) ) ) ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( inj_on_nat_nat @ F @ ( A @ I4 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_978_inj__on__UNION__chain,axiom,
! [I3: set_o,A: $o > set_nat,F: nat > nat] :
( ! [I4: $o,J2: $o] :
( ( member_o @ I4 @ I3 )
=> ( ( member_o @ J2 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A @ I4 ) @ ( A @ J2 ) )
| ( ord_less_eq_set_nat @ ( A @ J2 ) @ ( A @ I4 ) ) ) ) )
=> ( ! [I4: $o] :
( ( member_o @ I4 @ I3 )
=> ( inj_on_nat_nat @ F @ ( A @ I4 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_979_inj__on__UNION__chain,axiom,
! [I3: set_a,A: a > set_nat,F: nat > nat] :
( ! [I4: a,J2: a] :
( ( member_a @ I4 @ I3 )
=> ( ( member_a @ J2 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A @ I4 ) @ ( A @ J2 ) )
| ( ord_less_eq_set_nat @ ( A @ J2 ) @ ( A @ I4 ) ) ) ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( inj_on_nat_nat @ F @ ( A @ I4 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ A @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_980_inj__on__UNION__chain,axiom,
! [I3: set_complex,A: complex > set_nat,F: nat > nat] :
( ! [I4: complex,J2: complex] :
( ( member_complex @ I4 @ I3 )
=> ( ( member_complex @ J2 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A @ I4 ) @ ( A @ J2 ) )
| ( ord_less_eq_set_nat @ ( A @ J2 ) @ ( A @ I4 ) ) ) ) )
=> ( ! [I4: complex] :
( ( member_complex @ I4 @ I3 )
=> ( inj_on_nat_nat @ F @ ( A @ I4 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_6352962638927555131et_nat @ A @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_981_inj__on__UNION__chain,axiom,
! [I3: set_list_a,A: list_a > set_nat,F: nat > nat] :
( ! [I4: list_a,J2: list_a] :
( ( member_list_a @ I4 @ I3 )
=> ( ( member_list_a @ J2 @ I3 )
=> ( ( ord_less_eq_set_nat @ ( A @ I4 ) @ ( A @ J2 ) )
| ( ord_less_eq_set_nat @ ( A @ J2 ) @ ( A @ I4 ) ) ) ) )
=> ( ! [I4: list_a] :
( ( member_list_a @ I4 @ I3 )
=> ( inj_on_nat_nat @ F @ ( A @ I4 ) ) )
=> ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ A @ I3 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_982_SUP__UNION,axiom,
! [F: nat > $o,G2: nat > set_nat,A: set_nat] :
( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ A ) ) ) )
= ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [Y4: nat] : ( complete_Sup_Sup_o @ ( image_nat_o @ F @ ( G2 @ Y4 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_983_SUP__UNION,axiom,
! [F: nat > set_nat,G2: nat > set_nat,A: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ A ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y4: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ ( G2 @ Y4 ) ) )
@ A ) ) ) ).
% SUP_UNION
thf(fact_984_SUP__commute,axiom,
! [F: nat > nat > set_nat,B: set_nat,A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( F @ I2 ) @ B ) )
@ A ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [J: nat] :
( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [I2: nat] : ( F @ I2 @ J )
@ A ) )
@ B ) ) ) ).
% SUP_commute
thf(fact_985_UN__UN__flatten,axiom,
! [C2: nat > set_nat,B: nat > set_nat,A: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y4: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B @ Y4 ) ) )
@ A ) ) ) ).
% UN_UN_flatten
thf(fact_986_UN__E,axiom,
! [B3: $o,B: nat > set_o,A: set_nat] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ A ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_o @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_987_UN__E,axiom,
! [B3: $o,B: $o > set_o,A: set_o] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_o @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_988_UN__E,axiom,
! [B3: $o,B: a > set_o,A: set_a] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_a_set_o @ B @ A ) ) )
=> ~ ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_o @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_989_UN__E,axiom,
! [B3: $o,B: complex > set_o,A: set_complex] :
( ( member_o @ B3 @ ( comple90263536869209701_set_o @ ( image_complex_set_o @ B @ A ) ) )
=> ~ ! [X: complex] :
( ( member_complex @ X @ A )
=> ~ ( member_o @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_990_UN__E,axiom,
! [B3: a,B: nat > set_a,A: set_nat] :
( ( member_a @ B3 @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_a @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_991_UN__E,axiom,
! [B3: a,B: $o > set_a,A: set_o] :
( ( member_a @ B3 @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_a @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_992_UN__E,axiom,
! [B3: a,B: a > set_a,A: set_a] :
( ( member_a @ B3 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) )
=> ~ ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_a @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_993_UN__E,axiom,
! [B3: a,B: complex > set_a,A: set_complex] :
( ( member_a @ B3 @ ( comple2307003609928055243_set_a @ ( image_complex_set_a @ B @ A ) ) )
=> ~ ! [X: complex] :
( ( member_complex @ X @ A )
=> ~ ( member_a @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_994_UN__E,axiom,
! [B3: complex,B: nat > set_complex,A: set_nat] :
( ( member_complex @ B3 @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ B @ A ) ) )
=> ~ ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_complex @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_995_UN__E,axiom,
! [B3: complex,B: $o > set_complex,A: set_o] :
( ( member_complex @ B3 @ ( comple8424636186594484919omplex @ ( image_o_set_complex @ B @ A ) ) )
=> ~ ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_complex @ B3 @ ( B @ X ) ) ) ) ).
% UN_E
thf(fact_996_UN__extend__simps_I9_J,axiom,
! [C2: nat > set_nat,B: nat > set_nat,A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( B @ X3 ) ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ C2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_997_inj__on__image__Fpow,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( inj_on_nat_set_nat @ F @ A )
=> ( inj_on2776966659131765557et_nat @ ( image_nat_set_nat @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_998_inj__on__image__Fpow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_999_Sup__subset__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_1000_Sup__subset__mono,axiom,
! [A: set_set_complex,B: set_set_complex] :
( ( ord_le4750530260501030778omplex @ A @ B )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ A ) @ ( comple8424636186594484919omplex @ B ) ) ) ).
% Sup_subset_mono
thf(fact_1001_Sup__subset__mono,axiom,
! [A: set_set_list_a,B: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ B )
=> ( ord_le8861187494160871172list_a @ ( comple6928918032620976721list_a @ A ) @ ( comple6928918032620976721list_a @ B ) ) ) ).
% Sup_subset_mono
thf(fact_1002_Sup__subset__mono,axiom,
! [A: set_set_o,B: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A @ B )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ ( comple90263536869209701_set_o @ B ) ) ) ).
% Sup_subset_mono
thf(fact_1003_Sup__subset__mono,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_subset_mono
thf(fact_1004_Sup__subset__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_1005_SUP__eq,axiom,
! [A: set_nat,B: set_nat,F: nat > $o,G2: nat > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1006_SUP__eq,axiom,
! [A: set_nat,B: set_o,F: nat > $o,G2: $o > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1007_SUP__eq,axiom,
! [A: set_nat,B: set_a,F: nat > $o,G2: a > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1008_SUP__eq,axiom,
! [A: set_nat,B: set_complex,F: nat > $o,G2: complex > $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ? [X5: complex] :
( ( member_complex @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: complex] :
( ( member_complex @ J2 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_complex_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1009_SUP__eq,axiom,
! [A: set_o,B: set_nat,F: $o > $o,G2: nat > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1010_SUP__eq,axiom,
! [A: set_o,B: set_o,F: $o > $o,G2: $o > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1011_SUP__eq,axiom,
! [A: set_o,B: set_a,F: $o > $o,G2: a > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X5: a] :
( ( member_a @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: a] :
( ( member_a @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_a_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1012_SUP__eq,axiom,
! [A: set_o,B: set_complex,F: $o > $o,G2: complex > $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ? [X5: complex] :
( ( member_complex @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: complex] :
( ( member_complex @ J2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_complex_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1013_SUP__eq,axiom,
! [A: set_a,B: set_nat,F: a > $o,G2: nat > $o] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: nat] :
( ( member_nat @ J2 @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1014_SUP__eq,axiom,
! [A: set_a,B: set_o,F: a > $o,G2: $o > $o] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ? [X5: $o] :
( ( member_o @ X5 @ B )
& ( ord_less_eq_o @ ( F @ I4 ) @ ( G2 @ X5 ) ) ) )
=> ( ! [J2: $o] :
( ( member_o @ J2 @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A )
& ( ord_less_eq_o @ ( G2 @ J2 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_1015_finite__inverse__image__gen,axiom,
! [A: set_o,F: $o > $o,D: set_o] :
( ( finite_finite_o @ A )
=> ( ( inj_on_o_o @ F @ D )
=> ( finite_finite_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ D )
& ( member_o @ ( F @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1016_finite__inverse__image__gen,axiom,
! [A: set_o,F: a > $o,D: set_a] :
( ( finite_finite_o @ A )
=> ( ( inj_on_a_o @ F @ D )
=> ( finite_finite_a
@ ( collect_a
@ ^ [J: a] :
( ( member_a @ J @ D )
& ( member_o @ ( F @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1017_finite__inverse__image__gen,axiom,
! [A: set_o,F: nat > $o,D: set_nat] :
( ( finite_finite_o @ A )
=> ( ( inj_on_nat_o @ F @ D )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ D )
& ( member_o @ ( F @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1018_finite__inverse__image__gen,axiom,
! [A: set_o,F: complex > $o,D: set_complex] :
( ( finite_finite_o @ A )
=> ( ( inj_on_complex_o @ F @ D )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [J: complex] :
( ( member_complex @ J @ D )
& ( member_o @ ( F @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1019_finite__inverse__image__gen,axiom,
! [A: set_a,F: $o > a,D: set_o] :
( ( finite_finite_a @ A )
=> ( ( inj_on_o_a @ F @ D )
=> ( finite_finite_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ D )
& ( member_a @ ( F @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1020_finite__inverse__image__gen,axiom,
! [A: set_a,F: a > a,D: set_a] :
( ( finite_finite_a @ A )
=> ( ( inj_on_a_a @ F @ D )
=> ( finite_finite_a
@ ( collect_a
@ ^ [J: a] :
( ( member_a @ J @ D )
& ( member_a @ ( F @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1021_finite__inverse__image__gen,axiom,
! [A: set_a,F: nat > a,D: set_nat] :
( ( finite_finite_a @ A )
=> ( ( inj_on_nat_a @ F @ D )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ D )
& ( member_a @ ( F @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1022_finite__inverse__image__gen,axiom,
! [A: set_a,F: complex > a,D: set_complex] :
( ( finite_finite_a @ A )
=> ( ( inj_on_complex_a @ F @ D )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [J: complex] :
( ( member_complex @ J @ D )
& ( member_a @ ( F @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1023_finite__inverse__image__gen,axiom,
! [A: set_nat,F: $o > nat,D: set_o] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_o_nat @ F @ D )
=> ( finite_finite_o
@ ( collect_o
@ ^ [J: $o] :
( ( member_o @ J @ D )
& ( member_nat @ ( F @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1024_finite__inverse__image__gen,axiom,
! [A: set_nat,F: a > nat,D: set_a] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_a_nat @ F @ D )
=> ( finite_finite_a
@ ( collect_a
@ ^ [J: a] :
( ( member_a @ J @ D )
& ( member_nat @ ( F @ J ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_1025_Union__mono,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A ) @ ( comple2307003609928055243_set_a @ B ) ) ) ).
% Union_mono
thf(fact_1026_Union__mono,axiom,
! [A: set_set_complex,B: set_set_complex] :
( ( ord_le4750530260501030778omplex @ A @ B )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ A ) @ ( comple8424636186594484919omplex @ B ) ) ) ).
% Union_mono
thf(fact_1027_Union__mono,axiom,
! [A: set_set_list_a,B: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ B )
=> ( ord_le8861187494160871172list_a @ ( comple6928918032620976721list_a @ A ) @ ( comple6928918032620976721list_a @ B ) ) ) ).
% Union_mono
thf(fact_1028_Union__mono,axiom,
! [A: set_set_o,B: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A @ B )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ A ) @ ( comple90263536869209701_set_o @ B ) ) ) ).
% Union_mono
thf(fact_1029_Union__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_mono
thf(fact_1030_SUP__eqI,axiom,
! [A: set_nat,F: nat > $o,X2: $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y2: $o] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y2 ) )
=> ( ord_less_eq_o @ X2 @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1031_SUP__eqI,axiom,
! [A: set_o,F: $o > $o,X2: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y2: $o] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y2 ) )
=> ( ord_less_eq_o @ X2 @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1032_SUP__eqI,axiom,
! [A: set_a,F: a > $o,X2: $o] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y2: $o] :
( ! [I5: a] :
( ( member_a @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y2 ) )
=> ( ord_less_eq_o @ X2 @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1033_SUP__eqI,axiom,
! [A: set_complex,F: complex > $o,X2: $o] :
( ! [I4: complex] :
( ( member_complex @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y2: $o] :
( ! [I5: complex] :
( ( member_complex @ I5 @ A )
=> ( ord_less_eq_o @ ( F @ I5 ) @ Y2 ) )
=> ( ord_less_eq_o @ X2 @ Y2 ) )
=> ( ( complete_Sup_Sup_o @ ( image_complex_o @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1034_SUP__eqI,axiom,
! [A: set_nat,F: nat > set_a,X2: set_a] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y2: set_a] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_less_eq_set_a @ ( F @ I5 ) @ Y2 ) )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1035_SUP__eqI,axiom,
! [A: set_o,F: $o > set_a,X2: set_a] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y2: set_a] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_less_eq_set_a @ ( F @ I5 ) @ Y2 ) )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1036_SUP__eqI,axiom,
! [A: set_a,F: a > set_a,X2: set_a] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y2: set_a] :
( ! [I5: a] :
( ( member_a @ I5 @ A )
=> ( ord_less_eq_set_a @ ( F @ I5 ) @ Y2 ) )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1037_SUP__eqI,axiom,
! [A: set_complex,F: complex > set_a,X2: set_a] :
( ! [I4: complex] :
( ( member_complex @ I4 @ A )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y2: set_a] :
( ! [I5: complex] :
( ( member_complex @ I5 @ A )
=> ( ord_less_eq_set_a @ ( F @ I5 ) @ Y2 ) )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_complex_set_a @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1038_SUP__eqI,axiom,
! [A: set_nat,F: nat > set_complex,X2: set_complex] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_le211207098394363844omplex @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y2: set_complex] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A )
=> ( ord_le211207098394363844omplex @ ( F @ I5 ) @ Y2 ) )
=> ( ord_le211207098394363844omplex @ X2 @ Y2 ) )
=> ( ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1039_SUP__eqI,axiom,
! [A: set_o,F: $o > set_complex,X2: set_complex] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_le211207098394363844omplex @ ( F @ I4 ) @ X2 ) )
=> ( ! [Y2: set_complex] :
( ! [I5: $o] :
( ( member_o @ I5 @ A )
=> ( ord_le211207098394363844omplex @ ( F @ I5 ) @ Y2 ) )
=> ( ord_le211207098394363844omplex @ X2 @ Y2 ) )
=> ( ( comple8424636186594484919omplex @ ( image_o_set_complex @ F @ A ) )
= X2 ) ) ) ).
% SUP_eqI
thf(fact_1040_SUP__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat,G2: nat > set_nat] :
( ! [N3: nat] :
( ( member_nat @ N3 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ B ) ) ) ) ).
% SUP_mono
thf(fact_1041_SUP__mono,axiom,
! [A: set_o,B: set_nat,F: $o > set_nat,G2: nat > set_nat] :
( ! [N3: $o] :
( ( member_o @ N3 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ B ) ) ) ) ).
% SUP_mono
thf(fact_1042_SUP__mono,axiom,
! [A: set_a,B: set_nat,F: a > set_nat,G2: nat > set_nat] :
( ! [N3: a] :
( ( member_a @ N3 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ B ) ) ) ) ).
% SUP_mono
thf(fact_1043_SUP__mono,axiom,
! [A: set_complex,B: set_nat,F: complex > set_nat,G2: nat > set_nat] :
( ! [N3: complex] :
( ( member_complex @ N3 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_6352962638927555131et_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ B ) ) ) ) ).
% SUP_mono
thf(fact_1044_SUP__mono,axiom,
! [A: set_list_a,B: set_nat,F: list_a > set_nat,G2: nat > set_nat] :
( ! [N3: list_a] :
( ( member_list_a @ N3 @ A )
=> ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( G2 @ X5 ) ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ B ) ) ) ) ).
% SUP_mono
thf(fact_1045_SUP__least,axiom,
! [A: set_nat,F: nat > $o,U: $o] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_1046_SUP__least,axiom,
! [A: set_o,F: $o > $o,U: $o] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_1047_SUP__least,axiom,
! [A: set_a,F: a > $o,U: $o] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_1048_SUP__least,axiom,
! [A: set_complex,F: complex > $o,U: $o] :
( ! [I4: complex] :
( ( member_complex @ I4 @ A )
=> ( ord_less_eq_o @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_complex_o @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_1049_SUP__least,axiom,
! [A: set_nat,F: nat > set_a,U: set_a] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_1050_SUP__least,axiom,
! [A: set_o,F: $o > set_a,U: set_a] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_1051_SUP__least,axiom,
! [A: set_a,F: a > set_a,U: set_a] :
( ! [I4: a] :
( ( member_a @ I4 @ A )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_1052_SUP__least,axiom,
! [A: set_complex,F: complex > set_a,U: set_a] :
( ! [I4: complex] :
( ( member_complex @ I4 @ A )
=> ( ord_less_eq_set_a @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_complex_set_a @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_1053_SUP__least,axiom,
! [A: set_nat,F: nat > set_complex,U: set_complex] :
( ! [I4: nat] :
( ( member_nat @ I4 @ A )
=> ( ord_le211207098394363844omplex @ ( F @ I4 ) @ U ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_1054_SUP__least,axiom,
! [A: set_o,F: $o > set_complex,U: set_complex] :
( ! [I4: $o] :
( ( member_o @ I4 @ A )
=> ( ord_le211207098394363844omplex @ ( F @ I4 ) @ U ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ ( image_o_set_complex @ F @ A ) ) @ U ) ) ).
% SUP_least
thf(fact_1055_SUP__mono_H,axiom,
! [F: nat > set_nat,G2: nat > set_nat,A: set_nat] :
( ! [X: nat] : ( ord_less_eq_set_nat @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G2 @ A ) ) ) ) ).
% SUP_mono'
thf(fact_1056_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ I @ A )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1057_SUP__upper,axiom,
! [I: $o,A: set_o,F: $o > $o] :
( ( member_o @ I @ A )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1058_SUP__upper,axiom,
! [I: a,A: set_a,F: a > $o] :
( ( member_a @ I @ A )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1059_SUP__upper,axiom,
! [I: complex,A: set_complex,F: complex > $o] :
( ( member_complex @ I @ A )
=> ( ord_less_eq_o @ ( F @ I ) @ ( complete_Sup_Sup_o @ ( image_complex_o @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1060_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > set_a] :
( ( member_nat @ I @ A )
=> ( ord_less_eq_set_a @ ( F @ I ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1061_SUP__upper,axiom,
! [I: $o,A: set_o,F: $o > set_a] :
( ( member_o @ I @ A )
=> ( ord_less_eq_set_a @ ( F @ I ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1062_SUP__upper,axiom,
! [I: a,A: set_a,F: a > set_a] :
( ( member_a @ I @ A )
=> ( ord_less_eq_set_a @ ( F @ I ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1063_SUP__upper,axiom,
! [I: complex,A: set_complex,F: complex > set_a] :
( ( member_complex @ I @ A )
=> ( ord_less_eq_set_a @ ( F @ I ) @ ( comple2307003609928055243_set_a @ ( image_complex_set_a @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1064_SUP__upper,axiom,
! [I: nat,A: set_nat,F: nat > set_complex] :
( ( member_nat @ I @ A )
=> ( ord_le211207098394363844omplex @ ( F @ I ) @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1065_SUP__upper,axiom,
! [I: $o,A: set_o,F: $o > set_complex] :
( ( member_o @ I @ A )
=> ( ord_le211207098394363844omplex @ ( F @ I ) @ ( comple8424636186594484919omplex @ ( image_o_set_complex @ F @ A ) ) ) ) ).
% SUP_upper
thf(fact_1066_SUP__le__iff,axiom,
! [F: nat > set_nat,A: set_nat,U: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ U )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_1067_SUP__upper2,axiom,
! [I: nat,A: set_nat,U: $o,F: nat > $o] :
( ( member_nat @ I @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1068_SUP__upper2,axiom,
! [I: $o,A: set_o,U: $o,F: $o > $o] :
( ( member_o @ I @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1069_SUP__upper2,axiom,
! [I: a,A: set_a,U: $o,F: a > $o] :
( ( member_a @ I @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1070_SUP__upper2,axiom,
! [I: complex,A: set_complex,U: $o,F: complex > $o] :
( ( member_complex @ I @ A )
=> ( ( ord_less_eq_o @ U @ ( F @ I ) )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ ( image_complex_o @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1071_SUP__upper2,axiom,
! [I: nat,A: set_nat,U: set_a,F: nat > set_a] :
( ( member_nat @ I @ A )
=> ( ( ord_less_eq_set_a @ U @ ( F @ I ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1072_SUP__upper2,axiom,
! [I: $o,A: set_o,U: set_a,F: $o > set_a] :
( ( member_o @ I @ A )
=> ( ( ord_less_eq_set_a @ U @ ( F @ I ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1073_SUP__upper2,axiom,
! [I: a,A: set_a,U: set_a,F: a > set_a] :
( ( member_a @ I @ A )
=> ( ( ord_less_eq_set_a @ U @ ( F @ I ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1074_SUP__upper2,axiom,
! [I: complex,A: set_complex,U: set_a,F: complex > set_a] :
( ( member_complex @ I @ A )
=> ( ( ord_less_eq_set_a @ U @ ( F @ I ) )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ ( image_complex_set_a @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1075_SUP__upper2,axiom,
! [I: nat,A: set_nat,U: set_complex,F: nat > set_complex] :
( ( member_nat @ I @ A )
=> ( ( ord_le211207098394363844omplex @ U @ ( F @ I ) )
=> ( ord_le211207098394363844omplex @ U @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1076_SUP__upper2,axiom,
! [I: $o,A: set_o,U: set_complex,F: $o > set_complex] :
( ( member_o @ I @ A )
=> ( ( ord_le211207098394363844omplex @ U @ ( F @ I ) )
=> ( ord_le211207098394363844omplex @ U @ ( comple8424636186594484919omplex @ ( image_o_set_complex @ F @ A ) ) ) ) ) ).
% SUP_upper2
thf(fact_1077_UN__subset__iff,axiom,
! [A: nat > set_nat,I3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ I3 ) ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ I3 )
=> ( ord_less_eq_set_nat @ ( A @ X3 ) @ B ) ) ) ) ).
% UN_subset_iff
thf(fact_1078_UN__upper,axiom,
! [A2: nat,A: set_nat,B: nat > set_a] :
( ( member_nat @ A2 @ A )
=> ( ord_less_eq_set_a @ ( B @ A2 ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1079_UN__upper,axiom,
! [A2: $o,A: set_o,B: $o > set_a] :
( ( member_o @ A2 @ A )
=> ( ord_less_eq_set_a @ ( B @ A2 ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1080_UN__upper,axiom,
! [A2: a,A: set_a,B: a > set_a] :
( ( member_a @ A2 @ A )
=> ( ord_less_eq_set_a @ ( B @ A2 ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1081_UN__upper,axiom,
! [A2: complex,A: set_complex,B: complex > set_a] :
( ( member_complex @ A2 @ A )
=> ( ord_less_eq_set_a @ ( B @ A2 ) @ ( comple2307003609928055243_set_a @ ( image_complex_set_a @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1082_UN__upper,axiom,
! [A2: nat,A: set_nat,B: nat > set_complex] :
( ( member_nat @ A2 @ A )
=> ( ord_le211207098394363844omplex @ ( B @ A2 ) @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1083_UN__upper,axiom,
! [A2: $o,A: set_o,B: $o > set_complex] :
( ( member_o @ A2 @ A )
=> ( ord_le211207098394363844omplex @ ( B @ A2 ) @ ( comple8424636186594484919omplex @ ( image_o_set_complex @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1084_UN__upper,axiom,
! [A2: a,A: set_a,B: a > set_complex] :
( ( member_a @ A2 @ A )
=> ( ord_le211207098394363844omplex @ ( B @ A2 ) @ ( comple8424636186594484919omplex @ ( image_a_set_complex @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1085_UN__upper,axiom,
! [A2: complex,A: set_complex,B: complex > set_complex] :
( ( member_complex @ A2 @ A )
=> ( ord_le211207098394363844omplex @ ( B @ A2 ) @ ( comple8424636186594484919omplex @ ( image_5702600179605932057omplex @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1086_UN__upper,axiom,
! [A2: nat,A: set_nat,B: nat > set_o] :
( ( member_nat @ A2 @ A )
=> ( ord_less_eq_set_o @ ( B @ A2 ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1087_UN__upper,axiom,
! [A2: $o,A: set_o,B: $o > set_o] :
( ( member_o @ A2 @ A )
=> ( ord_less_eq_set_o @ ( B @ A2 ) @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ).
% UN_upper
thf(fact_1088_UN__least,axiom,
! [A: set_nat,B: nat > set_a,C2: set_a] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1089_UN__least,axiom,
! [A: set_o,B: $o > set_a,C2: set_a] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1090_UN__least,axiom,
! [A: set_a,B: a > set_a,C2: set_a] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1091_UN__least,axiom,
! [A: set_complex,B: complex > set_a,C2: set_a] :
( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ord_less_eq_set_a @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_complex_set_a @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1092_UN__least,axiom,
! [A: set_nat,B: nat > set_complex,C2: set_complex] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_le211207098394363844omplex @ ( B @ X ) @ C2 ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1093_UN__least,axiom,
! [A: set_o,B: $o > set_complex,C2: set_complex] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_le211207098394363844omplex @ ( B @ X ) @ C2 ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ ( image_o_set_complex @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1094_UN__least,axiom,
! [A: set_a,B: a > set_complex,C2: set_complex] :
( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le211207098394363844omplex @ ( B @ X ) @ C2 ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ ( image_a_set_complex @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1095_UN__least,axiom,
! [A: set_complex,B: complex > set_complex,C2: set_complex] :
( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ord_le211207098394363844omplex @ ( B @ X ) @ C2 ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ ( image_5702600179605932057omplex @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1096_UN__least,axiom,
! [A: set_nat,B: nat > set_o,C2: set_o] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_o @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1097_UN__least,axiom,
! [A: set_o,B: $o > set_o,C2: set_o] :
( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_o @ ( B @ X ) @ C2 ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) @ C2 ) ) ).
% UN_least
thf(fact_1098_UN__mono,axiom,
! [A: set_a,B: set_a,F: a > set_a,G2: a > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1099_UN__mono,axiom,
! [A: set_a,B: set_a,F: a > set_complex,G2: a > set_complex] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le211207098394363844omplex @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ ( image_a_set_complex @ F @ A ) ) @ ( comple8424636186594484919omplex @ ( image_a_set_complex @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1100_UN__mono,axiom,
! [A: set_a,B: set_a,F: a > set_o,G2: a > set_o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A ) ) @ ( comple90263536869209701_set_o @ ( image_a_set_o @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1101_UN__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_a,G2: nat > set_a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1102_UN__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_complex,G2: nat > set_complex] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_le211207098394363844omplex @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ F @ A ) ) @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1103_UN__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_o,G2: nat > set_o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A ) ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1104_UN__mono,axiom,
! [A: set_complex,B: set_complex,F: complex > set_a,G2: complex > set_a] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_complex_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_complex_set_a @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1105_UN__mono,axiom,
! [A: set_complex,B: set_complex,F: complex > set_complex,G2: complex > set_complex] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ord_le211207098394363844omplex @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ ( image_5702600179605932057omplex @ F @ A ) ) @ ( comple8424636186594484919omplex @ ( image_5702600179605932057omplex @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1106_UN__mono,axiom,
! [A: set_complex,B: set_complex,F: complex > set_o,G2: complex > set_o] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_complex_set_o @ F @ A ) ) @ ( comple90263536869209701_set_o @ ( image_complex_set_o @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1107_UN__mono,axiom,
! [A: set_o,B: set_o,F: $o > set_a,G2: $o > set_a] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_o_set_a @ G2 @ B ) ) ) ) ) ).
% UN_mono
thf(fact_1108_UN__extend__simps_I10_J,axiom,
! [B: set_nat > set_nat,F: nat > set_nat,A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A4: nat] : ( B @ ( F @ A4 ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ B @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1109_UN__extend__simps_I10_J,axiom,
! [B: nat > set_nat,F: nat > nat,A: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [A4: nat] : ( B @ ( F @ A4 ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1110_image__UN,axiom,
! [F: nat > set_nat,B: nat > set_nat,A: set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( comple548664676211718543et_nat
@ ( image_2194112158459175443et_nat
@ ^ [X3: nat] : ( image_nat_set_nat @ F @ ( B @ X3 ) )
@ A ) ) ) ).
% image_UN
thf(fact_1111_image__UN,axiom,
! [F: nat > nat,B: nat > set_nat,A: set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X3: nat] : ( image_nat_nat @ F @ ( B @ X3 ) )
@ A ) ) ) ).
% image_UN
thf(fact_1112_image__Union,axiom,
! [F: nat > set_nat,S: set_set_nat] :
( ( image_nat_set_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ ( image_nat_set_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_1113_image__Union,axiom,
! [F: nat > nat,S: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_1114_UN__extend__simps_I8_J,axiom,
! [B: nat > set_nat,A: set_set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [Y4: set_nat] : ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ Y4 ) )
@ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_1115_finite__imageD,axiom,
! [F: a > a,A: set_a] :
( ( finite_finite_a @ ( image_a_a @ F @ A ) )
=> ( ( inj_on_a_a @ F @ A )
=> ( finite_finite_a @ A ) ) ) ).
% finite_imageD
thf(fact_1116_finite__imageD,axiom,
! [F: nat > a,A: set_nat] :
( ( finite_finite_a @ ( image_nat_a @ F @ A ) )
=> ( ( inj_on_nat_a @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_1117_finite__imageD,axiom,
! [F: complex > a,A: set_complex] :
( ( finite_finite_a @ ( image_complex_a @ F @ A ) )
=> ( ( inj_on_complex_a @ F @ A )
=> ( finite3207457112153483333omplex @ A ) ) ) ).
% finite_imageD
thf(fact_1118_finite__imageD,axiom,
! [F: a > nat,A: set_a] :
( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
=> ( ( inj_on_a_nat @ F @ A )
=> ( finite_finite_a @ A ) ) ) ).
% finite_imageD
thf(fact_1119_finite__imageD,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_1120_finite__imageD,axiom,
! [F: complex > nat,A: set_complex] :
( ( finite_finite_nat @ ( image_complex_nat @ F @ A ) )
=> ( ( inj_on_complex_nat @ F @ A )
=> ( finite3207457112153483333omplex @ A ) ) ) ).
% finite_imageD
thf(fact_1121_finite__imageD,axiom,
! [F: a > complex,A: set_a] :
( ( finite3207457112153483333omplex @ ( image_a_complex @ F @ A ) )
=> ( ( inj_on_a_complex @ F @ A )
=> ( finite_finite_a @ A ) ) ) ).
% finite_imageD
thf(fact_1122_finite__imageD,axiom,
! [F: nat > complex,A: set_nat] :
( ( finite3207457112153483333omplex @ ( image_nat_complex @ F @ A ) )
=> ( ( inj_on_nat_complex @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_1123_finite__imageD,axiom,
! [F: complex > complex,A: set_complex] :
( ( finite3207457112153483333omplex @ ( image_1468599708987790691omplex @ F @ A ) )
=> ( ( inj_on2498852929715845839omplex @ F @ A )
=> ( finite3207457112153483333omplex @ A ) ) ) ).
% finite_imageD
thf(fact_1124_finite__imageD,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ F @ A ) )
=> ( ( inj_on_nat_set_nat @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_1125_finite__image__iff,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( finite_finite_a @ ( image_a_a @ F @ A ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_image_iff
thf(fact_1126_finite__image__iff,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( finite_finite_a @ ( image_nat_a @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_1127_finite__image__iff,axiom,
! [F: complex > a,A: set_complex] :
( ( inj_on_complex_a @ F @ A )
=> ( ( finite_finite_a @ ( image_complex_a @ F @ A ) )
= ( finite3207457112153483333omplex @ A ) ) ) ).
% finite_image_iff
thf(fact_1128_finite__image__iff,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_image_iff
thf(fact_1129_finite__image__iff,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_1130_finite__image__iff,axiom,
! [F: complex > nat,A: set_complex] :
( ( inj_on_complex_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_complex_nat @ F @ A ) )
= ( finite3207457112153483333omplex @ A ) ) ) ).
% finite_image_iff
thf(fact_1131_finite__image__iff,axiom,
! [F: a > complex,A: set_a] :
( ( inj_on_a_complex @ F @ A )
=> ( ( finite3207457112153483333omplex @ ( image_a_complex @ F @ A ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_image_iff
thf(fact_1132_finite__image__iff,axiom,
! [F: nat > complex,A: set_nat] :
( ( inj_on_nat_complex @ F @ A )
=> ( ( finite3207457112153483333omplex @ ( image_nat_complex @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_1133_finite__image__iff,axiom,
! [F: complex > complex,A: set_complex] :
( ( inj_on2498852929715845839omplex @ F @ A )
=> ( ( finite3207457112153483333omplex @ ( image_1468599708987790691omplex @ F @ A ) )
= ( finite3207457112153483333omplex @ A ) ) ) ).
% finite_image_iff
thf(fact_1134_finite__image__iff,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( inj_on_nat_set_nat @ F @ A )
=> ( ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_1135_card__image,axiom,
! [F: a > a,A: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( finite_card_a @ ( image_a_a @ F @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_1136_card__image,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( finite_card_a @ ( image_nat_a @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_1137_card__image,axiom,
! [F: complex > a,A: set_complex] :
( ( inj_on_complex_a @ F @ A )
=> ( ( finite_card_a @ ( image_complex_a @ F @ A ) )
= ( finite_card_complex @ A ) ) ) ).
% card_image
thf(fact_1138_card__image,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_a_nat @ F @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_1139_card__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_1140_card__image,axiom,
! [F: complex > nat,A: set_complex] :
( ( inj_on_complex_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_complex_nat @ F @ A ) )
= ( finite_card_complex @ A ) ) ) ).
% card_image
thf(fact_1141_card__image,axiom,
! [F: a > complex,A: set_a] :
( ( inj_on_a_complex @ F @ A )
=> ( ( finite_card_complex @ ( image_a_complex @ F @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_1142_card__image,axiom,
! [F: nat > complex,A: set_nat] :
( ( inj_on_nat_complex @ F @ A )
=> ( ( finite_card_complex @ ( image_nat_complex @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_1143_card__image,axiom,
! [F: complex > complex,A: set_complex] :
( ( inj_on2498852929715845839omplex @ F @ A )
=> ( ( finite_card_complex @ ( image_1468599708987790691omplex @ F @ A ) )
= ( finite_card_complex @ A ) ) ) ).
% card_image
thf(fact_1144_card__image,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( inj_on_nat_set_nat @ F @ A )
=> ( ( finite_card_set_nat @ ( image_nat_set_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_1145_card__Union__le__sum__card,axiom,
! [U3: set_set_list_a] : ( ord_less_eq_nat @ ( finite_card_list_a @ ( comple6928918032620976721list_a @ U3 ) ) @ ( groups5993734322560061562_a_nat @ finite_card_list_a @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_1146_card__Union__le__sum__card,axiom,
! [U3: set_set_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( comple2307003609928055243_set_a @ U3 ) ) @ ( groups6141743369313575924_a_nat @ finite_card_a @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_1147_card__Union__le__sum__card,axiom,
! [U3: set_set_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( comple8424636186594484919omplex @ U3 ) ) @ ( groups8758837469787661168ex_nat @ finite_card_complex @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_1148_card__Union__le__sum__card,axiom,
! [U3: set_set_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( comple8404747032580312297st_nat @ U3 ) ) @ ( groups7315335787803791778at_nat @ finite_card_list_nat @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_1149_card__Union__le__sum__card,axiom,
! [U3: set_set_list_complex] : ( ord_less_eq_nat @ ( finite5120063068150530198omplex @ ( comple2136024025076962759omplex @ U3 ) ) @ ( groups6516131157293929088ex_nat @ finite5120063068150530198omplex @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_1150_card__Union__le__sum__card,axiom,
! [U3: set_set_list_list_a] : ( ord_less_eq_nat @ ( finite9134805042761151410list_a @ ( comple6939822128159878743list_a @ U3 ) ) @ ( groups2871842722561159296_a_nat @ finite9134805042761151410list_a @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_1151_card__Union__le__sum__card,axiom,
! [U3: set_set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( comple7399068483239264473et_nat @ U3 ) ) @ ( groups8294997508430121362at_nat @ finite_card_nat @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_1152_SUP__subset__mono,axiom,
! [A: set_a,B: set_a,F: a > $o,G2: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_a_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_a_o @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1153_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > $o,G2: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_nat_o @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1154_SUP__subset__mono,axiom,
! [A: set_complex,B: set_complex,F: complex > $o,G2: complex > $o] :
( ( ord_le211207098394363844omplex @ A @ B )
=> ( ! [X: complex] :
( ( member_complex @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_complex_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_complex_o @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1155_SUP__subset__mono,axiom,
! [A: set_o,B: set_o,F: $o > $o,G2: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ ( complete_Sup_Sup_o @ ( image_o_o @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1156_SUP__subset__mono,axiom,
! [A: set_a,B: set_a,F: a > set_a,G2: a > set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1157_SUP__subset__mono,axiom,
! [A: set_a,B: set_a,F: a > set_complex,G2: a > set_complex] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_le211207098394363844omplex @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ ( image_a_set_complex @ F @ A ) ) @ ( comple8424636186594484919omplex @ ( image_a_set_complex @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1158_SUP__subset__mono,axiom,
! [A: set_a,B: set_a,F: a > set_o,G2: a > set_o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X: a] :
( ( member_a @ X @ A )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_a_set_o @ F @ A ) ) @ ( comple90263536869209701_set_o @ ( image_a_set_o @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1159_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_a,G2: nat > set_a] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_a @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ F @ A ) ) @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1160_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_complex,G2: nat > set_complex] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_le211207098394363844omplex @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_le211207098394363844omplex @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ F @ A ) ) @ ( comple8424636186594484919omplex @ ( image_6594795319511438139omplex @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1161_SUP__subset__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_o,G2: nat > set_o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ord_less_eq_set_o @ ( F @ X ) @ ( G2 @ X ) ) )
=> ( ord_less_eq_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ F @ A ) ) @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ G2 @ B ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_1162_finite__UnionD,axiom,
! [A: set_set_a] :
( ( finite_finite_a @ ( comple2307003609928055243_set_a @ A ) )
=> ( finite_finite_set_a @ A ) ) ).
% finite_UnionD
thf(fact_1163_finite__UnionD,axiom,
! [A: set_set_complex] :
( ( finite3207457112153483333omplex @ ( comple8424636186594484919omplex @ A ) )
=> ( finite6551019134538273531omplex @ A ) ) ).
% finite_UnionD
thf(fact_1164_finite__UnionD,axiom,
! [A: set_set_list_a] :
( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ A ) )
=> ( finite5282473924520328461list_a @ A ) ) ).
% finite_UnionD
thf(fact_1165_finite__UnionD,axiom,
! [A: set_set_nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) )
=> ( finite1152437895449049373et_nat @ A ) ) ).
% finite_UnionD
thf(fact_1166_sum_Oimage__eq,axiom,
! [G2: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ G2 @ A )
=> ( ( groups3542108847815614940at_nat
@ ^ [X3: nat] : X3
@ ( image_nat_nat @ G2 @ A ) )
= ( groups3542108847815614940at_nat @ G2 @ A ) ) ) ).
% sum.image_eq
thf(fact_1167_subseqs__refl,axiom,
! [Xs2: list_a] : ( member_list_a @ Xs2 @ ( set_list_a2 @ ( subseqs_a @ Xs2 ) ) ) ).
% subseqs_refl
thf(fact_1168_endo__inj__surj,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ A )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_1169_endo__inj__surj,axiom,
! [A: set_complex,F: complex > complex] :
( ( finite3207457112153483333omplex @ A )
=> ( ( ord_le211207098394363844omplex @ ( image_1468599708987790691omplex @ F @ A ) @ A )
=> ( ( inj_on2498852929715845839omplex @ F @ A )
=> ( ( image_1468599708987790691omplex @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_1170_endo__inj__surj,axiom,
! [A: set_list_a,F: list_a > list_a] :
( ( finite_finite_list_a @ A )
=> ( ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A ) @ A )
=> ( ( inj_on_list_a_list_a @ F @ A )
=> ( ( image_list_a_list_a @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_1171_endo__inj__surj,axiom,
! [A: set_o,F: $o > $o] :
( ( finite_finite_o @ A )
=> ( ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ A )
=> ( ( inj_on_o_o @ F @ A )
=> ( ( image_o_o @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_1172_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1173_enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( member_nat @ S2 @ S )
=> ? [N3: nat] :
( ( infini8530281810654367211te_nat @ S @ N3 )
= S2 ) ) ) ).
% enumerate_Ex
thf(fact_1174_le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).
% le_enumerate
thf(fact_1175_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_1176_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1177_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_nat @ X @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1178_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1179_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M2: nat] :
( ( ord_less_nat @ K @ M2 )
=> ? [N6: nat] :
( ( ord_less_nat @ M2 @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1180_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1181_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1182_finite__le__enumerate,axiom,
! [S: set_nat,N: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_1183_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1184_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1185_nat__power__less__imp__less,axiom,
! [I: nat,M5: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M5 ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M5 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1186_finite__nth__roots,axiom,
! [N: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_1187_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1188_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1189_GreatestI__ex__nat,axiom,
! [P: nat > $o,B3: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_1190_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_1191_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_1192_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B3 ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1193_nat__le__linear,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
| ( ord_less_eq_nat @ N @ M5 ) ) ).
% nat_le_linear
thf(fact_1194_le__antisym,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( ord_less_eq_nat @ N @ M5 )
=> ( M5 = N ) ) ) ).
% le_antisym
thf(fact_1195_eq__imp__le,axiom,
! [M5: nat,N: nat] :
( ( M5 = N )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% eq_imp_le
thf(fact_1196_le__trans,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ J3 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1197_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1198_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1199_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1200_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1201_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1202_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1203_less__imp__le__nat,axiom,
! [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% less_imp_le_nat
thf(fact_1204_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1205_less__or__eq__imp__le,axiom,
! [M5: nat,N: nat] :
( ( ( ord_less_nat @ M5 @ N )
| ( M5 = N ) )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1206_le__neq__implies__less,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( M5 != N )
=> ( ord_less_nat @ M5 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1207_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J3: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1208_card__nth__roots,axiom,
! [C: complex,N: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z5: complex] :
( ( power_power_complex @ Z5 @ N )
= C ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_1209_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1210_UN__atMost__UNIV,axiom,
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
= top_top_set_nat ) ).
% UN_atMost_UNIV
thf(fact_1211_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1212_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_1213_range__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat )
= S ) ) ).
% range_enumerate
thf(fact_1214_mult__le__cancel2,axiom,
! [M5: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M5 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M5 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1215_le__cube,axiom,
! [M5: nat] : ( ord_less_eq_nat @ M5 @ ( times_times_nat @ M5 @ ( times_times_nat @ M5 @ M5 ) ) ) ).
% le_cube
thf(fact_1216_le__square,axiom,
! [M5: nat] : ( ord_less_eq_nat @ M5 @ ( times_times_nat @ M5 @ M5 ) ) ).
% le_square
thf(fact_1217_mult__le__mono,axiom,
! [I: nat,J3: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J3 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1218_mult__le__mono1,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J3 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1219_mult__le__mono2,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J3 ) ) ) ).
% mult_le_mono2
thf(fact_1220_nat__mult__le__cancel__disj,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M5 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1221_nat__mult__le__cancel1,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M5 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1222_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M5: nat] :
( ! [K4: nat] :
( ( ord_less_nat @ N @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K4 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K4 ) ) )
=> ( P @ M5 ) ) ) ).
% nat_descend_induct
thf(fact_1223_div__mult__mono,axiom,
! [A2: nat,D2: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ D2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ ( times_times_nat @ A2 @ B3 ) @ D2 ) @ B3 ) ) ) ).
% div_mult_mono
thf(fact_1224_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M5: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M5 @ ( divide_divide_nat @ N @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M5 @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1225_div__le__dividend,axiom,
! [M5: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M5 @ N ) @ M5 ) ).
% div_le_dividend
thf(fact_1226_div__le__mono,axiom,
! [M5: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M5 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1227_div__times__less__eq__dividend,axiom,
! [M5: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M5 @ N ) @ N ) @ M5 ) ).
% div_times_less_eq_dividend
thf(fact_1228_times__div__less__eq__dividend,axiom,
! [N: nat,M5: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M5 @ N ) ) @ M5 ) ).
% times_div_less_eq_dividend
thf(fact_1229_div__le__mono2,axiom,
! [M5: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M5 ) ) ) ) ).
% div_le_mono2
thf(fact_1230_div__greater__zero__iff,axiom,
! [M5: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M5 @ N ) )
= ( ( ord_less_eq_nat @ N @ M5 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1231_Suc__le__mono,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M5 ) )
= ( ord_less_eq_nat @ N @ M5 ) ) ).
% Suc_le_mono
thf(fact_1232_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M5: nat] :
( ( ( power_power_nat @ X2 @ M5 )
= ( suc @ zero_zero_nat ) )
= ( ( M5 = zero_zero_nat )
| ( X2
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1233_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1234_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_1235_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_1236_one__le__mult__iff,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M5 @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M5 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1237_Suc__div__le__mono,axiom,
! [M5: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M5 @ N ) @ ( divide_divide_nat @ ( suc @ M5 ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1238_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1239_transitive__stepwise__le,axiom,
! [M5: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ! [X: nat] : ( R2 @ X @ X )
=> ( ! [X: nat,Y2: nat,Z6: nat] :
( ( R2 @ X @ Y2 )
=> ( ( R2 @ Y2 @ Z6 )
=> ( R2 @ X @ Z6 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M5 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1240_nat__induct__at__least,axiom,
! [M5: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( P @ M5 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1241_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_eq_nat @ ( suc @ M6 ) @ N3 )
=> ( P @ M6 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1242_not__less__eq__eq,axiom,
! [M5: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M5 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M5 ) ) ).
% not_less_eq_eq
thf(fact_1243_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1244_le__Suc__eq,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M5 @ N )
| ( M5
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1245_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M2: nat] :
( M7
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_1246_le__SucI,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_nat @ M5 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1247_le__SucE,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M5 @ N )
=> ( M5
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1248_Suc__leD,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% Suc_leD
thf(fact_1249_card_Ocomp__fun__commute__on,axiom,
( ( comp_nat_nat_nat @ suc @ suc )
= ( comp_nat_nat_nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_1250_Suc__leI,axiom,
! [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( ord_less_eq_nat @ ( suc @ M5 ) @ N ) ) ).
% Suc_leI
thf(fact_1251_Suc__le__eq,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N )
= ( ord_less_nat @ M5 @ N ) ) ).
% Suc_le_eq
thf(fact_1252_dec__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J3 ) ) ) ) ).
% dec_induct
thf(fact_1253_inc__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J3 )
=> ( ( P @ J3 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J3 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1254_Suc__le__lessD,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N )
=> ( ord_less_nat @ M5 @ N ) ) ).
% Suc_le_lessD
thf(fact_1255_le__less__Suc__eq,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M5 ) )
= ( N = M5 ) ) ) ).
% le_less_Suc_eq
thf(fact_1256_less__Suc__eq__le,axiom,
! [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M5 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1257_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1258_le__imp__less__Suc,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_nat @ M5 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1259_inj__Suc,axiom,
! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).
% inj_Suc
thf(fact_1260_Suc__mult__le__cancel1,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M5 ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M5 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1261_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1262_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1263_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_1264_card__less,axiom,
! [M: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1265_card__less__Suc,axiom,
! [M: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M )
& ( ord_less_nat @ K3 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1266_card__less__Suc2,axiom,
! [M: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M )
& ( ord_less_nat @ K3 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1267_div__nat__eqI,axiom,
! [N: nat,Q2: nat,M5: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M5 )
=> ( ( ord_less_nat @ M5 @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
=> ( ( divide_divide_nat @ M5 @ N )
= Q2 ) ) ) ).
% div_nat_eqI
thf(fact_1268_split__div_H,axiom,
! [P: nat > $o,M5: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M5 @ N ) )
= ( ( ( N = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M5 )
& ( ord_less_nat @ M5 @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
& ( P @ Q3 ) ) ) ) ).
% split_div'
thf(fact_1269_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_1270_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= ( suc
@ ( ord_Least_nat
@ ^ [M3: nat] : ( P @ ( suc @ M3 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_1271_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1272_infinite__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ? [R3: nat > nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ R3 )
& ! [N6: nat] : ( member_nat @ ( R3 @ N6 ) @ S ) ) ) ).
% infinite_enumerate
thf(fact_1273_strict__mono__enumerate,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ ( infini8530281810654367211te_nat @ S ) ) ) ).
% strict_mono_enumerate
thf(fact_1274_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ N ) ) ) ).
% mono_times_nat
% Conjectures (1)
thf(conj_0,conjecture,
( ( finite_card_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ a2 )
& ( ( size_size_list_a @ Xs )
= n ) ) ) )
= ( power_power_nat @ ( finite_card_a @ a2 ) @ n ) ) ).
%------------------------------------------------------------------------------